Download Motivic Homotopy Theory

Ecole Polytechnique Fédérale de Lausanne, Suisse
Faculty of Mathematics
Spring 2012
Motivic Homotopy Theory
Bogdan Gheorghe
Advisor: Kathryn Hess Bellwald
Abstract
The goal of this project is to introduce motivic homotopy theory, which is a homotopy
theory for schemes. Given a small category of k-schemes Sch/k, the Yoneda embedding embeds it fully faithfully in the category of simplicial presheaves [Sch/k op , sSet], which admits
(several) model structures inherited from sSet. Unfortunately, these model structures do
not preserve the colimits of Sch/k. The game is to refine these model structures until they
reflect the ’geometry of schemes’ and resemble standard homotopy theories.
Contents
Contents
1
Introduction
3
1 Prerequisites
1.1 Prerequisites from Category Theory . . . . . . . . .
1.2 Enriched Category Theory . . . . . . . . . . . . . . .
1.2.1 Monoidal categories . . . . . . . . . . . . . .
1.2.2 Enriched categories . . . . . . . . . . . . . . .
1.3 Presheaves and Sheaves on Grothendieck Topologies
1.3.1 Grothendieck topologies . . . . . . . . . . . .
1.3.2 Sheaves on Grothendieck sites . . . . . . . . .
1.4 Simplicial and Cosimplicial Objects . . . . . . . . . .
1.4.1 (Co)simplicial objects and (co)skeletons . . .
1.4.2 Augmented simplicial objects . . . . . . . . .
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32
2 Additional Structures on Model Categories
2.1 Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 A few categorical prerequisites . . . . . . . . . . . . . . . . . . . . . .
2.1.2 The definition of a model category and examples . . . . . . . . . . . .
2.1.3 The construction of the homotopy category . . . . . . . . . . . . . . .
2.1.4 Functors between model categories . . . . . . . . . . . . . . . . . . . .
2.1.5 Cofibrantely generated model categories and the small object argument
2.1.6 Cellular and combinatorial model categories . . . . . . . . . . . . . . .
2.1.7 Proper model categories . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Simplicial Model Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Simplicial categories . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Simplicial model categories . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Localization of Model Categories . . . . . . . . . . . . . . . . . . . . . . . . .
34
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71
3 Motivic Homotopy Theory
3.1 Global Model Structures on Simplicial Presheaves . . . . . . . . . . . . . . . .
3.1.1 The global projective model structure . . . . . . . . . . . . . . . . . .
3.1.2 Comparison between injective and projective global models . . . . . .
80
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88
1
3.2
3.3
3.4
3.5
Universal Model Categories and Small Presentations . .
3.2.1 Universal cocompletion . . . . . . . . . . . . . .
3.2.2 Universal homotopy cocompletion . . . . . . . .
3.2.3 Small presentations . . . . . . . . . . . . . . . . .
Local Model Structures on Simplicial Presheaves . . . .
3.3.1 Hypercompletion . . . . . . . . . . . . . . . . . .
3.3.2 Characterization of cofibrant and fibrant objects
The Category of (Nisnevich) Motivic Spaces . . . . . . .
3.4.1 The category of motivic spaces . . . . . . . . . .
3.4.2 Homotopy theory on motivic spaces . . . . . . .
3.4.3 The category of Nisnevich motivic spaces . . . .
Unstable Motivic Homotopy Theory . . . . . . . . . . .
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126
Index
132
Bibliography
134
2
Introduction
The idea of the subject came at the end of a lecture in algebraic K-theory given by my advisor
for this project, Professor Kathryn Hess Bellwald. I would like to take this opportunity to
thank her for all her help and support for this work. Without any doubt I learned far more
mathematics during this semester under her supervision than during any other. I am also
thankful and in debt to Marc Hoyois, for answering so nicely to my emails and sharing his
algebro-geometric intuition.
The initial motivation for this project was to define motivic cohomology, in order to
understand how it helps in the computation of the K-theory of the integers. Unfortunately,
for lack of time, neither of these two topics was covered. Therefore, I would like in this
introduction to explain the road to what was my leitmotiv for almost all this work, the
definition of motivic cohomology.
Motivic cohomology is a cohomology theory for schemes, conjectured in the 60’s by
Alexander Grothendieck. I highly recommend the very readable exposition text [Kah07]
related to the subject. This cohomology theory was supposed to satisfy axioms making
it into a universal cohomology for schemes, in the categorical sense that any other such
cohomology should factor through it. To better understand what this cohomology theory is
and how it arises, we need to (re)define more precisely what a cohomology theory is.
For topological spaces, a cohomology theory is a defined as a (contravariant) functor
from Top to, say graded abelian groups, satisfying the Eilenberg–Steenrod axioms. There
are many attempts to formulate such axioms for schemes, but none of them really broke
through, as did the Eilenberg-Steenrod axioms in topology. As a general idea, the important
properties of a cohomology theory are the following three.
(1) Being invariant with respect to a reasonable notion of homotopy : this is the role of such
a theory, to give (computable) invariants up to homotopy, which is a weaker version of
an isomorphism;
(2) Commuting with coproducts (or filtered colimits) : the cohomology of a disjoint union
is computable in terms of the cohomology of the components;
(3) A local-to-global principle (excision) : the cohomology of a compatible local data should
compute the entire cohomology.
For topological spaces, the last axiom is defined in the sense that, for example, the coho`
mology of the pushout Y = X X0 X 0
3
X0
X
X0
pY
is determined by the cohomology of its pieces X, X 0 and X0 .
If we start with some category of schemes, there are a few problems. First, what will be
the notion of a weak equivalence of schemes and a homotopy between morphisms of schemes
? The notion of a weak equivalence between the underlying topological spaces is certainly
not enough since the structural sheaf is also part of the structure of a scheme. Moreover,
given a homotopy theory of schemes, how to construct cohomology theories ?
An answer to the first question is given by the machinery invented by Quillen, the
model categories, introduced in [Qui67]. A model structure on a category M is the data of
three classes of maps, the cofibrations C , the fibrations F and the weak equivalences W
satisfying some axioms. The weak equivalences give raise to a homotopy theory on M, and
to a homotopy category Ho(M) which is the localization M[W −1 ] of the initial category
forcing all the weak equivalences to be isomorphisms. The additional data of cofibrations
and fibrations have no influence on the homotopy category, but they ensure its existence and
allow an explicit construction. It is usually hard to understand and construct (arbitrary)
localizations of categories; one of the greatest strength of model categories is to explicitly
construct the localization M[W −1 ], as a sort of quotient of M. Model categories have
proved to be very useful in algebraic topology and homological algebra, giving a common
framework in which the study of objects up to (chain) homotopy is possible. One of their
first applications outside these fields, is to the definition of A1 -homotopy theory (also called
motivic homotopy theory) in algebraic geometry, which is exactly the model structure on
schemes that we look for. The resulting model category is called the (unstable) category
of motivic spaces, and we will denote it by MS . In fact, it is the development of the
A1 -homotopy theory (and MS ) and the further definition of motivic cohomology that led
Vladimir Voevodksy to the award of a Fields Medal in 2002.
Before explaining in more details how this homotopy theory for schemes is defined, let’s
see how this leads to cohomology theories. The key fact is to use a variant of Brown’s
representability theorem, which roughly says that, in the case of topological spaces, every
cohomology theory is represented by an object in the homotopy category of spectra1 . Mimicking this property, one could hope that similarly, by inverting some suspension functor
in the category of motivic spaces MS , this would give a category where the cohomology
theories for schemes live. It turns out that this construction actually leads to the stable
motivic category, giving access to cohomology theories for schemes.
So how is this model structure on schemes constructed, leading to a homotopy theory
of schemes ? I will here briefly describe this construction, and refer to the (non-published)
article [Dug] for a further discussion.
1
The category of spectra of topological spaces is the category in which the suspension functor has been
inverted.
4
More concretely, let’s endow the category Sch/k of schemes of finite type over an algebraically closed field k with a homotopy theory. The first observation is that from a
categorical point of view, the category Sch/k is intractable since it does not contain all
colimits. A good (universal) way to solve this problem is by fully faithfully embedding it
by the Yoneda embedding, into its category of presheaves. That is, embedding it in the
category of functors Sch/k op GGA Set.
However, the category of sets is not meant for homotopy theory. In a certain sense,
there is not enough room for homotopic deformations in this category, so we should instead
consider functors from the category of schemes into sSet, the category of simplicial sets. The
category of functors Sch/k op GGA sSet is called the category of simplicial presheaves, and is
the starting point for defining a model structure. As any category of functors (or diagrams),
it inherits object-wise most of the structure of the target category sSet. In particular, the
natural homotopy theory of sSet, which is equivalent to the homotopy theory of Top, is
inherited by the category of simplicial presheaves. As we will see, this homotopy theory
does not reflect the geometry of schemes, in the sense that some colimits are not computed
as geometric intuition would expect them to be. Here, the geometric intuition is interpreted
by the underlying topological space of the scheme. So the problem is that the underlying
topological space of a colimit of schemes is not the same as the colimit of the underlying
topological spaces.
A way to deal with this problem is by adding an additional structure to the category
Sch/k of schemes, and quotienting in some way by it, in the category of simplicial presheaves.
This is encoded in a Grothendieck topology on Sch/k, which gives the notion of a covering.
By specifying which family of morphisms will be an ’open cover’ in the category of schemes,
there is a way, called (Bousfield) localization, to force these colimits to be preserved in the
category of simplicial presheaves.
Another problem is that the category of simplicial presheaves lacks the crucial notion of
an interval I. However, the affine line A1 (k) may play this role, and again by localization,
we may force it to homotopically act as an interval, whatever that role may be.
The question now is, did we extract enough geometric properties from Top in order to
have an interesting homotopy theory of schemes ? The article [Dug01c] from Dugger explains
that all this construction may be enough. Indeed, by starting with a more geometric category
than Sch/k, we can apply the same construction and see what comes out of it. Roughly
speaking, the construction is the following, starting with any category C.
(1)
(2)
(3)
(4)
Cocomplete it by taking presheaves on it;
Extend this to simplicial presheaves to add a homotopy theory;
Endow C with a Grothendieck topology and localize for having the geometric colimits;
Choose a reasonable interval in C, and force it to homotopically behave like one.
It turns out that, by starting with the simplicial category ∆ with interval ∆[1], we get back
the homotopy theory of simplicial sets, and by starting with the category of real manifolds
ManR with interval R, we get back the homotopy theory of real manifolds ! Moreover, by
skipping the last step, in the case of ∆, the homotopy theory is almost the homotopy theory
of simplicial sets, except that it does not know that ∆[n] must be contractible, and similarly
5
for ManR we get the homotopy theory of manifolds, without R being contractible. That
is, the last step is necessary and there are reasons to believe that this machinery gives a
reasonable homotopy theory for schemes.
Starting with some category of schemes, the output of this machinery is called the
unstable motivic category, and this is as far as this project goes. In order to define motivic
cohomology for schemes, the unstable motivic category may be stabilized by inverting two
suspension functors, and then picking the right object in the homotopy category of the stable
motivic category that represents motivic cohomology. Even though this is explained in a
few lines here, complications arise since there are two suspension functors : one associated
to the sphere in simplicial sets, and one associated to the sphere in the category of schemes
(the multiplicative group), see for example [DLØ+ 07] for more explanations.
Let’s now provide an overview of the mathematical content of this project. For each
section, we list the important definitions and results, and give the reference it is taken from.
When no reference is given, it usually means that this is taken from the internet, mostly
ncatlab.org.
The first chapter contains various prerequisites. The reader is assumed familiar to be
with the basic notions of category theory, simplicial sets, and homotopy groups of topological
spaces. The first section contains a brief review of Kan extensions [Mac71]. The second
section defines monoidal categories and then categories enriched over monoidal categories
[Bor94a]. Our prototype of a monoidal category is the category of simplicial sets with the
categorical product, and later the categories of functors in simplicial sets. In the third
section we define sites to be categories endowed with a Grothendieck topology, as well as
the notion of (pre)sheaves on sites [Art62]. We explore the relation between presheaves and
sheaves by means of the sheafification adjoint. In the last section we define (augmented)
simplicial objects in general categories, which is a convenient language that will be used
later.
The second chapter is dedicated to the study of model categories. The usual references,
from which all the chapter is taken, are [Hov99] for a general approach and an emphasis on
the homotopy category, [Hir03] for an emphasis on localizations but also a huge amount of
the general theory, [GJ09] for an emphasis on simplicial examples, and [DS95] for a general
introduction. In the first section we first give the definition and first properties of a model
category, as well as many examples. We then give the usual construction of its homotopy
category, and define the notion of functors between model categories, Quillen adjunctions.
We then define cofibrantly generated model categories, which are given with a much smaller
amount of data than a usual model category. We explaine the small object argument, which
is a generic argument that ’constructs’ model structures from two well-chosen generating
sets. We then define cellular and combinatorial model categories, which are model categories
that are cofibrantly generated, in a stronger sense. Finally, we define properness in model
categories, which is an extra useful property that model categories may enjoy. The second
section treats simplicial model categories, which are model categories that are enriched over
sSet, where the enrichment is required to be compatible with the model structure. Such
categories are very useful as they carry a natural notion of a simplicial mapping space
between any two objects. Finally the third section is devoted to the heavy machinery of
6
localization, from [Hir03]. This section is important as it will be used many times later on.
The third and last chapter finally treats motivic homotopy theory. In the first section,
we explain how to endow a category of C-diagrams in a model category M, with a model
structure primarily coming object-wise from M. Most of the proof for the model structure
is taken from [Hov99]. We then specialize it to C-diagrams in sSet and characterize cofibrant and fibrant objects. In the second section we give the first step towards a homotopy
theory of schemes, by showing first how to universally cocomplete a category (by formally
adding colimits), and then how to universally homotopy cocomplete it (by formally adding
homotopy colimits) [Dug01c]. Starting with a category of schemes, these categories may
now be endowed with a model structure from the first section. Finally, we define the notion
of small presentation of a model category. In the third section, we take care of the third step
mentioned above, and localize with respect to the Grothendieck topology. This is done by a
more general procedure called hypercompletion [DHI04]. In the fourth section, we specialize
the construction to categories of schemes. We first start by studying the categorical properties of such categories, then endow it with the first model structure. Then, we define the
Nisnevich topology, which is the Grothendieck topology that will be used for A1 -homotopy
theory. All the work done previously, allows us to formally localize the model structure with
respect to this topology. Finally in the last section, we do the last step and localize with
respect to the interval A1 . This gives the unstable motivic category that defines a homotopy
theory for schemes.
7
1. Prerequisites
1.1
Prerequisites from Category Theory
We will assume familiarities with the basic notions of category theory such as (locally small)
categories, functors, natural transformations, all kinds of (small) limits and colimits, completeness and cocompletness, adjunctions and categories of functors. There are many good
introductions to the subject, for example the book of Borceux [Bor94a] or the standard
[Mac71]. The notion of simplicial sets as well as homotopy groups of topological spaces is
also recommended.
For a category C, we will denote its class of objects by Ob(C) or simply by C. All the
concrete categories are assumed to be locally small, i.e., the hom-sets between any two
objects are actual sets (elements of the category Set of sets). Most of the usual categories
are indeed locally small, even tough in some constructions, we may leave the world of locally
small categories. We recall the Yoneda lemma and Kan extensions.
Lemma 1.1.1 (Yoneda lemma). Let C be a (locally small) category, and let C ∈ C be an
object. For any functor F : C GGA Set, there is a (natural) bijection
Nat(C(C, −), F ) ∼
= F (C)
∈ Set,
between the natural transformations C(C, −) =⇒ F and the elements of F (C).
Similarly, for any functor F : C op GGA Set, there is a (natural) bijection
Nat(C(−, C), F ) ∼
= F (C)
∈ Set,
between the natural transformations C(−, C) =⇒ F and the elements of F (C).
Given two functors
i
C
A
F
M,
it is sometimes useful to be able to extend the functor F to a functor G : A GGA M. It
may not be possible to extend it strictly, such that G ◦ i = F , but only up to a natural
transformation, either G ◦ i =⇒ F or in the other direction F GGGA G ◦ i. When such
8
CHAPTER 1. PREREQUISITES
extensions exist, there are two (extremal) universal ones that are called the left and right
Kan extension of F along i. In particular, such extensions exist when i is a fully faithful
embedding (if C is a subcategory of A for example) and if M is complete and cocomplete.
In this case, the natural transformations G ◦ i =⇒ F and F =⇒ G ◦ i are in fact natural
isomorphisms.
Definition (Left and right Kan extensions). A left Kan extension of F along i is a functor
ε
L : A GGA M with a natural transformation F =⇒ L ◦ i, that is universal, in the sense
described below.
Dually, a right Kan extension of F along i is a functor R : A GGA M with a natural
ε
transformation R ◦ i =⇒ F , that is universal, in the sense described below.
For a left Kan extension, the universality means that for any other functor L0 : A GGA M
ε0
µ
with a natural transformation F =⇒ L0 ◦ i, there is a unique natural transformation L =⇒ L0
such that the composite
µ∗id
ε
F =⇒ L ◦ i =⇒ L0 ◦ i
is equal to ε0 . Dually, for a right extension, the universality means that for any other
ε0
functor R0 : A GGA M with a natural transformation R0 ◦ i =⇒ F , there is a unique natural
µ
transformation R0 =⇒ R such that the composite
µ∗id
ε
R0 ◦ i =⇒ R ◦ i =⇒ F
is equal to ε0 .
The functor i : C GGA A induces a functor by precomposition
i∗ : MA GGA MC .
Observe that the functor categories MA and MC may not be locally small categories if
either C or A are not small, but in our application the categories C and A will be small.
The best possible scenario is if i∗ has a left adjoint or a right adjoint. Indeed, if there exists
a left adjoint Li
A
∗
GGA
Li : MC DG
G ⊥G M : i ,
then the left Kan extension of F along i is the functor Li (F ) ∈ MA with the unit of the
adjunction F =⇒ Li (F ) ◦ i as natural transformation. Dually, if there is a right adjoint
C
GGA
i∗ : MA DG
G ⊥ G M : Ri ,
then the right Kan extension of F along i is the functor Ri (F ) with the counit of the
adjunction Ri (F ) ◦ i =⇒ F as natural transformation. It is important to emphasize the fact
that it is not necessary that i∗ admits a left or right adjoint in order for a functor F to
admit a left or right Kan extension. However, we have the following theorem that gives the
existence of such adjoints.
Theorem 1.1.2. If C is small and M is complete, then i∗ admits a right adjoint Ri
C
GGA
i∗ : MA DG
G ⊥ G M : Ri ,
9
CHAPTER 1. PREREQUISITES
and therefore any functor F : C GGA M admits a right Kan extension along i.
Dually, if M is cocomplete, then i∗ admits a left adjoint Li
A
∗
GGA
Li : MC DG
G⊥G M : i ,
and therefore any functor F : C GGA M admits a left Kan extension along i.
Proof. This is Corollary 2 in Section X.3 in [Mac71].
Corollary 1.1.3. In addition, if the functor i : C ,GGA A is full and faithful, then the unit
F =⇒ Li (F ) ◦ i and the counit Ri (F ) ◦ i =⇒ F are natural isomorphisms.
Proof. This is Corollary 3 in Section X.3 in [Mac71].
For example, if we set C = ∆ the simplex category, consider the diagram
∆
i
sSet
ρ
Top,
where
i : ∆ ,GGA sSet : [n] G
[ GA ∆(−, n)
is the Yoneda embedding, and where we define
ρ : ∆ ,GGA Top : [n] G
[ GA ∆n .
Since ∆ is small and Top is cocomplete, ρ admits a left Kan extension along i
∆
i
sSet
Re
ρ
Top,
called the geometric realization. Moreover, this functor admits a right adjoint
GGA
Re : sSet DG
G ⊥ G Top : Sing,
called the singular functor. See Proposition 3.2.1 for a generalization of this construction.
10
CHAPTER 1. PREREQUISITES
1.2
Enriched Category Theory
Recall that we assumed that all our categories are locally small. In this section, we will define
the notion of an enriched category, which may be seen as a generalization of an ordinary
category. It often happens that in a category C, the hom-sets C(A, B) have more structure
than just being sets. In fact, most of the standard categories admit extra structure on their
hom-sets, for example
(1) the hom-set Top(X, Y ) between two topological spaces can be endowed with the compactopen topology;
(2) the hom-set Ab(G, H) between two abelian groups is an abelian group under the addition of homomorphisms;
(3) the hom-set R Mod(M, N ) between two (left) R-modules is also an abelian group under
the addition of R-linear maps;
(4) the hom-set VecF (V, W ) between two F-vector spaces is again an F-vector space;
and the list can continue for many other categories. A category C is said to be enriched over
another category V 1 if the hom-sets are objects of V, denoted by C(A, B) ∈ V, and if there
are composition maps
C(A, B) × C(B, C) GGA C(A, C)
∈ V,
satisfying an associativity and unit axiom. Implicitly, here we made use of the categorical
product in V, by computing C(A, B) × C(B, C). However, the categorical product does not
always have good properties (it is not always in adjunction with Hom for example). We
will now define the categories with good products, on which we will be able to enrich other
categories.
1.2.1
Monoidal categories
We will enrich categories over what is called a monoidal category, that is, a category equipped
with a bifunctor −⊗− : V ×V GGA V that will play the role of the above product of the homobjects. This introduction to monoidal categories and enriched categories is essentially from
Section 6 in the book [Bor94b] of Borceux. Another good reference with many examples is
given in Chapter 4 of [Hov99].
Definition (Monoidal category). A monoidal category V is a category V together with
• a bifunctor − ⊗ − : V × V GGA V called the tensor product;
• a distinguished object I ∈ V called the unit;
• for every triple of objects A, B, C ∈ V a natural isomorphism
∼
=
αABC : (A ⊗ B) ⊗ C GGA A ⊗ (B ⊗ C),
called the associator of A, B and C;
1
A more rigorous definition is given later in 1.2.2.
11
CHAPTER 1. PREREQUISITES
• for every object A ∈ V a natural isomorphism
∼
=
lA : I ⊗ A GGA A,
called the left unit of A;
• for every object A ∈ V a natural isomorphism
∼
=
rA : A ⊗ I GGA A,
called the right unit of A;
such that for any objects A, B, C, D ∈ V, the following two diagrams of associativity coherence and unit coherence commute.
αA⊗B,C,D
((A ⊗ B) ⊗ C) ⊗ D
(A ⊗ B) ⊗ (C ⊗ D)
(A ⊗ I) ⊗ B
αAIB
A ⊗ (I ⊗ B)
αABC ⊗ id
αA,B,C⊗D
(A ⊗ (B ⊗ C)) ⊗ D
rA ⊗ id
id ⊗lB
αA,B⊗C,D ⊗ id
A ⊗ ((B ⊗ C) ⊗ D)
A ⊗ (B ⊗ (C ⊗ D))
id ⊗αBCD
A ⊗ B.
Observe that all the arrows in both diagrams are isomorphisms in V. We usually refer
to such a monoidal category by (V, ⊗, I) or even simply by V if the context is clear, letting
understood the product, the unit (unique up to isomorphism), and the associators and unit
isomorphisms.
Definition (Symmetric). A monoidal category (V, ⊗, I) is called symmetric if in addition
there are natural isomorphisms
∼
=
sAB : A ⊗ B GGA B ⊗ A,
such that the following diagrams of associativity coherence, unit coherence and symmetry
coherence commute.
(A ⊗ B) ⊗ C
sAB ⊗ id
αABC
αBAC
A ⊗ (B ⊗ C)
B ⊗ (A ⊗ C)
sA⊗B,C
(B ⊗ C) ⊗ A
(B ⊗ A) ⊗ C
id ⊗sAC
αBCA
12
B ⊗ (C ⊗ A)
CHAPTER 1. PREREQUISITES
sAI
A⊗I
ra
A
I ⊗A
la
sAB
B⊗A
A⊗B
sBA
A ⊗ B.
Again, all arrows in those three diagrams are isomorphisms in V. All the monoidal
categories that we will consider will indeed be symmetric. Before giving some examples,
let’s give a last useful property that we would like monoidal categories to satisfy.
Definition (Closed symmetric monoidal category). A symmetric monoidal category V is
said to be closed, if for all objects A ∈ V, the endofunctor − ⊗ A has a right adjoint. This
right adjoint is usually denoted by (−)A .
Observe that it follows that A ⊗ − also has a right adjoint, since these two functors are
naturally isomorphic by
∼
=
s−,A : − ⊗A =⇒ A ⊗ −.
The right adjoint is denoted by (−)A because it really corresponds to an exponentiation, as
is showed in the following examples.
Example 1.1 (Monoidal categories).
(1) The most basic example of a monoidal category is the category Set of sets with the
cartesian product × as tensor product and the unit being a singleton {∗}. Moreover, it is
clearly symmetric since A × B ∼
= B × A naturally, and it is moreover closed. The adjoint
functor of − × B is the covariant hom-functor (−)B = Set(B, −), and the adjunction
is sometimes called the exponential law
∼ Set(A, Set(B, C)).
Set(A × B, C) =
(2) Let V be a category that admits all finite products. By choosing a terminal object
∗ and a fixed product A × B for all objects A, B ∈ V, the category V is a monoidal
category with the categorical product × as tensor product and ∗ as unit. This product
is also symmetric since A × B ∼
= B × A (by a unique isomorphism) by definition of the
categorical product. If the monoidal structure is closed, the right adjoint (−)B gives
the notion of an internal-hom, also denoted by AB = V(B, A) ∈ V.
(3) In particular, the category Top of topological spaces is symmetric monoidal with the
cartesian product ×, the unit ∗ and the natural isomorphisms X ×Y ∼
= Y ×X. Moreover,
a candidate for a right adjoint to −×Y is the exponential functor that gives an internalhom
(−)Y = Top(Y, −) : Top GGA Top : Z G
[ GA Top(Y, Z),
where Top(Y, Z) is endowed with the compact-open topology. However, the isomorphism
Top(X × Y, Z) ∼
= Top(X, Top(Y, Z))
13
CHAPTER 1. PREREQUISITES
only holds when Y is locally compact and Hausdorff, and in fact it turns out that
Top is not closed. This is a reason why the category of topological spaces is sometimes restricted to the full subcategory of compactly generated spaces, which is a closed
symmetric monoidal category under the cartesian product.
(4) Similarly, the category sSet of simplicial sets is symmetric monoidal with the cartesian
product and the unit {∗}∗ . Moreover, it is a closed symmetric monoidal category, where
the right adjoint of − × Y• is given by the internal-hom
(−)Y• = sSet(Y• , −) : sSet GGA sSet : Z• G
[ GA sSet(Y• , Z• )
which is given by sSet(Y• , Z• )n := sSet(Y• × ∆[n], Z• ).
(5) The category Ab of abelian groups is a monoidal category with tensor product ⊗ and
unit Z. Moreover, it is symmetric since A ⊗ B ∼
= B ⊗ A naturally, and it is closed by
the tensor-hom adjunction
∼ Ab(A, Ab(B, C)),
Ab(A ⊗ B, C) =
where Ab(B, C) is given the structure of an abelian group under pointwise addition.
(6) More generally, if R is a commutative ring the category R Mod of R-modules is a
monoidal category with tensor product ⊗R and unit R. It is symmetric since M ⊗R N ∼
=
N ⊗R M naturally, and it is closed by the same tensor-hom adjunction
R Mod(M
∼ R Mod(M, R Mod(N, L)),
⊗R N, L) =
where R Mod(N, L) is given the structure of an R-module under pointwise addition and
pointwise multiplication by R.
(7) An example that we will often use is the induced monoidal structure on a category of
diagrams. If C is a small category and (V, ⊗, I) is a monoidal category, the category of
functors [C, V] admits a monoidal structure with pointwise tensor product
F ⊗ G(C) := F (C) ⊗ G(C),
and where the unit is given by the constant functor
I˜: C GGA V : C G
[ GA I(C) = I.
Moreover, it is symmetric if V is, and in addition closed if V is. This example is treated
in more details in Example 1.2 at page 19, or when V = sSet at the beginning of Section
3.4.
(8) The pointed versions Top∗ and sSet∗ are also symmetric monoidal categories under the
smash product ∧, and the unit given by ∗+ , where we consider the adjunctions
GGA
(−)+ : Top DG
G ⊥ G Top∗ : U
and
GGA
(−)+ : sSet DG
G ⊥ G sSet∗ : U,
where (−)+ adds a disjoint base point, i.e., X+ := X ∗, and U is the forgetful functor.
They are both symmetric, and sSet∗ is again closed with a similar adjoint (−)Y• .
`
14
CHAPTER 1. PREREQUISITES
Most of these categories admit an internal-hom bifunctor V op × V GGA V, that will be
denoted by V(−, −), as already did in the previous examples. As we will see, it turns out
that any closed symmetric monoidal category admits such an internal-hom functor. The
covariant functor V(I, −) : V GGA Set represented by the unit I is called the underlying set
functor, as it gives back the underlying set of the objects in many situations. For example in
Set since Set(∗, X) ∼
= X, in Top since Top(∗, X) ∼
= X, in R Mod since R Mod(R, M ) ∼
= M,
and so on.
Let’s now consider a closed symmetric monoidal category (V, ⊗, I), where the adjunctions
are denoted by
GGA
− ⊗ A : V DG
G ⊥ G V : V(A, −).
The unit of this adjunction gives morphisms B GGA V(A, B ⊗ A), which specializes to
I GGA V(A, I ⊗ A) ∼
= V(A, A),
by letting B = I. It turns out that in the previous examples when I = ∗, the image of this
morphism gives back the identity morphism on A. For example in Set, this morphism is
{∗} GGA Set(A, A) : ∗ G
[ GA idA .
The counit of the adjunction gives evaluation morphisms
evAB : V(A, B) ⊗ A GGA B,
which for example in Set are
Set(A, B) × A GGA B : (f, a) G
[ GA f (a).
These evaluation morphisms induce a composition
cABC : V(A, B) ⊗ V(B, C) GGA V(A, C),
which is given as the adjoint morphism of the composite
evAB ⊗id
evBC
∼ V(B, C)⊗B GGA
V(A, B)⊗V(B, C)⊗A ∼
C.
= V(A, B)⊗A⊗V(B, C) GGA B ⊗V(B, C) =
Proposition 1.2.1. On a symmetric monoidal closed category (V, ⊗, I), there is a bifunctor
V(−, −) : V op × V GGA V : (A, B) G
[ GA V(A, B),
whose postcomposite with the underlying set functor V(I, −) gives the hom-functor
V(−, −) : V op × V GGA Set : (A, B) G
[ GA V(A, B).
Proof. We first have that V(A, −) is a functor, which is the right adjoint to − ⊗ A. For the
other variable, for any morphism f : A GGA A0 , define
V(f, B) : V(A0 , B) GGA V(A, B)
15
CHAPTER 1. PREREQUISITES
to be the corresponding morphism by adjunction, to the composite
evA0 B
id ⊗f
V(A0 , B) ⊗ A GGA V(A0 , B) ⊗ A0 GGA B,
and thehe functoriality of the bifunctor V(−, −) follows.
For the second part, the composite gives
(A, B) G
[ GA V(A, B) G
[ GA V(I, V(A, B)),
which is isomorphic in Set to V(A, B) since
V(I, V(A, B)) ∼
= V(I ⊗ A, B) ∼
= V(A, B).
1.2.2
Enriched categories
In this projet, the main purpose for defining monoidal categories is to further define categories enriched over them2 .
Definition (Enriched category). Let (V, ⊗, I) be a monoidal category. A V-category or a
category enriched over V is
• a collection of objects Ob(C);
• for every pair of objects A, B ∈ Ob(C), an object C(A, B) ∈ V;
• for every triple of objects A, B, C ∈ Ob(C), a composition morphism
C(A, B) ⊗ C(B, C) GGA C(A, C)
∈ V;
• for every object A ∈ Ob(C), a unit morphism
I GGA C(A, A)
∈ V;
such that the associativity axiom and the unit axiom hold, i.e., the following two diagrams
commute
(C(A, B) ⊗ C(B, C)) ⊗ C(C, D)
cABC ⊗ id
C(A, C) ⊗ C(C, D)
αC(A,B),C(B,C),C(C,D)
cACD
C(A, B) ⊗ (C(B, C) ⊗ C(C, D))
id ⊗cBCD
C(A, B) ⊗ C(B, D)
cABD
2
C(A, D)
Another important aspect of monoidal categories is to define algebras over them. Hovey is more focused
on this purpose in [Hov99].
16
CHAPTER 1. PREREQUISITES
I ⊗ C(A, B)
lC(A,B)
uA ⊗ id
C(A, A) ⊗ C(A, B)
lC(A,B)
C(A, B)
id ⊗uB
id
cAAB
C(A, B) ⊗ I
C(A, B)
cABB
C(A, B) ⊗ C(B, B).
Since the axioms are always similar, a shorter way to state and remember them would
be to say that
• the two ways of getting from (C(A, B) ⊗ C(B, C)) ⊗ C(C, D) to C(A, D) should be the
same (associativity);
• the two ways of getting from I ⊗ C(A, B) to C(A, B) are the same (left unit);
• the two ways of getting from C(A, B) ⊗ I to C(A, B) are the same (right unit).
The ordinary theory of locally small categories can be seen as the theory of categories
enriched over Set. In a similar way, a category with all finite limits and colimits is a preabelian category if and only if it is enriched over the category Ab of abelian groups. We
will mostly be interested in two types of enriched categories
• categories enriched over spaces, usually over sSet;
• monoidal categories enriched over themselves.
Proposition 1.2.2. Let (V, ⊗, I) be a closed symmetric monoidal category. Then V can
naturally be enriched over itself.
Proof. Since it is closed, the tensor product − ⊗ A has a right adjoint
V(A, −) : V GGA V,
and Proposition 1.2.1 shows the existence of a bifunctor
V(−, −) : V op × V GGA V.
The unit of the adjunction, specialized at I gives the unit morphism
I GGA V(A, A),
and the counit (evaluation) of the adjunction, applied two times gives a composition morphism
V(A, B) ⊗ V(B, C) GGA V(A, C).
It remains to check the commutativity of the diagrams, which is a fastidious but a straightforward checking.
17
CHAPTER 1. PREREQUISITES
Observe that in the definition of a V-category C, there is no mention of C being a
category. Applying Proposition 1.2.1 and Proposition 1.2.2 we get that starting with a
closed symmetric monoidal category V, if we enrich it over iself with internal-hom V(A, B),
applying the functor
V(I, −) : V GGA Set
gives back the hom V(A, B) of the underlying category. There is a more general statement
which says that given a V-category C, there is an underlying category, also denoted by C.
The general construction, that appears in Chapter 6 of [Bor94b] is as follows. We first define
V-functors between V-categories and V-natural transformations between them, and the small
V-categories with these notions form a 2-category, denoted by V-Cat. All these definitions
are natural and are what we expect them to be. We can then define the V-representables
and we get an enriched Yoneda lemma. Moreover, for every monoidal category V, there is
a forget 2-functor3
U : V-Cat GGA Cat
that admits a left adjoint and really behaves like a forget functor. It satisfies U (C)(A, B) =
V(I, C(A, B)) for any V-category C, and in particular applying it to V gives U (V)(A, B) =
V(A, B), as expected. This functor is in fact a functor V-Cat GGA Set-Cat, which is the
identity on objects. In view of these remarks, we may often abuse notation and identify an
enriched category C with the underlying category, also denoted by C.
The last notion that we need from enriched category theory is the notion of a tensor
and cotensor. Given a category C enriched over a symmetric closed monoidal category V,
this provides a way to compute tensor products C ⊗ V ∈ C for an object C ∈ C and V ∈ V,
as well as the adjoint operation, the power V C ∈ C.
Definition (Tensor and cotensor). Let V be a symmetric monoidal closed category, C be a
V-category and pick two objects C ∈ C and V ∈ V.
• The tensor of V and C, if it exists, is an object denoted by V ⊗ C ∈ C together with
isomorphisms in V
C(V ⊗ C, C 0 ) ∼
= V(V, C(C, C 0 ))
natural in C 0 ∈ C. We say that C is tensored (over V) when all tensors V ⊗ C ∈ C exist.
• The cotensor of V and C, if it exists, is an object denoted by C V ∈ C together with
isomorphisms in V
C(C 0 , C V ) ∼
= V(V, C(C 0 , C))
natural in C 0 ∈ C. We say that C is cotensored (over V) when all cotensors C V ∈ C
exist.
If C is tensored and cotensored over V, it follows that there are natural isomorphisms
C(V ⊗ C 0 , C) ∼
= C(C 0 , C V )
∈ V,
that can be interpreted as an enriched adjunction.
3
Roughly speaking, this forget 2-functor is a usual functor that has the additional structure of sending
V-natural transformations to natural transformations.
18
CHAPTER 1. PREREQUISITES
Example 1.2 (Enriched categories).
(1) Any locally small category C is enriched over (Set, ×, {∗}). The enriched hom-sets
are the usual hom-sets C(A, B) ∈ Set, the composition is the usual one and the unit
morphism is {∗} GGA C(A, A) : ∗ G
[ GA idA . Moreover, if C has coproducts, the tensor
of S ∈ Set and A ∈ C must satisfy
C(S ⊗ A, B) ∼
= Set(S, C(A, B)) ∼
=
Y
C(A, B) ∼
=C
S
!
a
A, B ,
S
and if C has products, the cotensor of S ∈ Set and A ∈ C must satisfy
C(B, A ) ∼
= Set(S, C(B, A)) ∼
=
S
Y
C(B, A) ∼
= C B,
S
!
Y
A .
S
Therefore, if C has products and coproducts, then C is tensored and cotensored where
`
the tensor can be given by the copower S A and the cotensor can be given by the
Q
power S A. For this reason, a tensor is sometimes called a copower and a cotensor is
called a power.
(2) Let V be a symmetric monoidal closed category. Then the V-category V is both tensored
and cotensored, with tensor V ⊗ A and cotensor AV . In particular, all the previous
symmetric monoidal closed categories Set, sSet, sSet∗ , compactly generated spaces,
Ab, R Mod (for R a commutative ring), . . . are tensored and cotensored over themselves.
(3) Consider the closed symmetric monoidal category Ab under the tensor product ⊗ and
unit Z. For a non-necessary commutative ring R, the category ModR of right Rmodules can be enriched over Ab under pointwise addition of R-linear maps. However,
under no additional conditions on R (such as commutativity for example), the abelian
groups ModR (M, N ) are not R-modules, in general. This category is tensored and
cotensored over Ab where, for two objects M ∈ ModR and A ∈ Ab, the tensor is the
right R-module A ⊗Z M ∈ ModR satisfying
ModR (A ⊗Z M, N ) ∼
= Ab(A, ModR (M, N ))
∈ Ab,
and the cotensor of A and M satisfies
ModR (N, M A ) ∼
= Ab(A, ModR (N, M ))
∈ Ab.
It is given by the right R-module which is Ab(A, M ) endowed with the right action
Ab(A, M ) × R GGA Ab(A, M ) : (f, r) G
[ GA f ∗ r,
where (f ∗ r)(a) := f (a) · r ∈ M .
(4) An example of an enriched category that we will often use is a category of C-diagrams in
a monoidal category V, that can be enriched either over itself or over V. More precisely,
let C be a small category and consider the category of functors [C op , sSet] that we will
denote by M := [C op , sSet]. The only reason of taking C op instead of C is because
the category M of functors is called the category of simplicial presheaves on C and is of
19
CHAPTER 1. PREREQUISITES
important use in algebraic geometry. Consider the symmetric closed monoidal structure
on sSet given by the categorical (cartesian) product ×, the constant simplicial set {∗}∗
as unit and the internal-homs are given by sSet(X• , Y• ) ∈ sSet where
sSet(X• , Y• )n := sSet(X• × ∆[n], Y• )
∈ Set.
This monoidal structure induces two different structures on M. First, it induces a
monoidal structure on M and therefore induces a structure of M-enriched category
on itself. Second, we can also see M as enriched over sSet, and it is tensored and
cotensored.
The induce monoidal structure is the pointwise one. For F, G ∈ M, their product is
(F × G)(C) := F (C) × G(C) ∈ sSet,
the unit is the constant simplicial presheaf ∗ : C G
[ GA ∗ that sends any object C ∈ C to
the unit of ∗ of sSet. Moreover, it is symmetric and closed since sSet is (as already
pointed out in Examples 1.1.
For any simplicial set K• consider the simplicial presheaf K̃• : C G
[ GA K• which is
constant on objects. We can now define the tensor and cotensor of F ∈ M with
K• ∈ sSet by
K• ⊗ F := K̃• × F
and
F K• := F K̃• , i.e.,
f
(−)
taking the operation after the embedding sSet ,GGA M. The category M := [C op , sSet]
of simplicial presheaves is enriched over sSet by defining MsSet (F, G) ∈ sSet to be
˜
M(F × ∆[−],
G), i.e.,
˜
MsSet (F, G)n := M(F × ∆[n],
G)
∈ Set.
It follows that the enrichement is tensored and cotensored by the above formulas.
A follow-up of this section appears in section 2.2 where we first explain in more details
what means to be enriched over sSet, and then explore the compatibility between a model
structure and an enriched structure over simplicial sets.
1.3
Presheaves and Sheaves on Grothendieck Topologies
In this section we will add another type of structure on a category, a Grothendieck topology.
The role of a Grothendieck topology on a category is to give accessible the notion of what
an open cover is. The goal is to mimic the natural notion of open cover that appears in
the category Top of topological spaces, or in the category ManR of real manifolds. This
can be seen as adding a geometric flavour in the category. The notion of an open cover
{Uα ,GGA X} allows a transition from local informations on Uα to global informations on
X, and the presheaves that respect these transitions will be called sheaves.
20
CHAPTER 1. PREREQUISITES
1.3.1
Grothendieck topologies
Definition (Grothendieck (pre)topology, site). Let C be a category that admits all pullbacks. A Grothendieck pretopology on C is an assignment to each object X ∈ C of a collection
of families of morphisms {Uα GGA X}α ⊆ Mor(C) called covering families (of X), satisfying
the axioms
∼
=
∼
=
(1) any isomorphism U GGA X gives a covering family of X with one morphism {U GGA
X};
(2) for any covering family {Uα GGA X}α of X and any morphism Y GGA X, the projections Uα ×X Y GGA Y from the pullback squares
Uα ×X Y
Y
y
Uα
X
form a covering family {Uα ×X Y GGA Y }α of Y ;
(3) for any covering family {Uα GGA X}α of X and every covering families {Vα,β GGA Uα }β
for each Uα , the composite {Vα,β GGA Uα GGA X}α,β is again a (finer) covering family
of X.
A category C with the additional structure of a Grothendieck topology is called a site.
If in addition the category C is small, then it is called a small site.
The data of all these covering families defines a pretopology on C, and generates a unique
topology on C. Even though there can be different pretopologies that generate the same
topology, it is not really restrictive to only work with pretopologies. The important point
is that sheaves only depends on the topology, so it is enough to consider a pretopology that
generates the wanted topology. It is therefore usually convenient to work with pretopologies
instead of topologies.
A Grothendieck topology is a similar definition as a pretopology, that is given in term
of covering sieves instead of covering families. A covering sieve can be seen as the closure
under precomposition of morphisms in the category, of a covering family, and is therefore a
much ’bigger’ data than a covering family.
As the definition of a topology is more technical and gives no additional insight in what
these topologies are, we will only illustrate what a sieve intuitively is. If a covering family
of X can be seen as a 1-iterated covering containing only morphisms Uα GGA X (as on the
left picture), a sieve can be seen as the closure by precomposition of a covering family (as
on the right picture)
21
CHAPTER 1. PREREQUISITES
Uα
Uβ
..
.
Vα,β
X
Uα
···
Wα,β,γ
Uβ
···
..
.
..
.
···
Wβ,γ,δ
X
Vβ,γ
···
and contains the closure by precomposition of a given covering family. Of course, the
diagram of a covering sieve (of X here) is much more complicated than this drawing, but
this already gives an idea that it is more convenient to work with covering families. Working
with pretopologies can be compared with working with a basis of a vector space, the choice is
not unqiue but they all generate the same space by some closure operations. We will usually
identify a pretopology with the topology it generates and therefore work with a topology
that is defined in terms of covering families. The only axiom of a (pre)topology that may
sound mysterious is the second axiom with the pullback property, which is explained in the
following examples.
Example 1.3 (Grothendieck topologies, sites).
(1) The prototype example of a small site is the category top(X) for a topological space
X. The category top(X) is the poset under inclusion ⊆ of the open subsets of X, i.e.,
it is defined by
• Objects : inclusions U ,GGA X;
• Morphisms : inclusions U ,GGA V between the underlying open subsets of X, such
that the triangle commutes
X
U
V.
The notion of a covering family in top(X) is the usual one, that is, a family of morphisms
S
{Uα ,GGA W }α in the category top(X) is a covering family if and only if Uα covers
W . In top(X), colimits are unions and limits are intersections. More precisely, the
’pullback over X’
22
CHAPTER 1. PREREQUISITES
V
U
W
X
∼
=
is the intersection U ∩V ,GGA X ∈ top(X). Since the only isomorphisms are U ,GGA U ,
the first axiom of a pretopology is satisfied. The second axiom can be restated as if
{Uα ,GGA W }α is a covering family of W and V ,GGA W is another morphism in
top(X), then the morphisms in the pullbacks
Uα ×W V = Uα ∩ V
V
Uα
W
give a covering family {Uα ∩ V ,GGA V }α of V . In words, this says that the intersection
with V of a covering of W gives of covering of V . The last axiom says that if {Uα ,GGA
W }α is a covering of W and if {Vα,β ,GGA Uα }β are coverings of each Uα , then the
composite {Vα,β ,GGA Uα ,GGA W }α,β is a covering of W . This can be seen as a finer
covering that refines the covering by Uα ’s.
(2) The large version of the small sites top(X) is the big site Top of all topological spaces.
A collection {Uα GGA X}α is defined to be a covering family in Top if and only if it is
so in top(X). In particular the morphisms Uα ,GGA X are required to be injections.
(3) We will be mostly interested in topologies on categories of schemes. Let k be an algebraically closed field, and let Sch/k be the category of schemes over k, i.e., morphisms
of schemes (X, OX ) GGA (Spec k, Ok ). We will usually denote a scheme (X, OX ) by the
underlying topological space X. As we did with topological spaces, let’s fix a k-scheme
S ∈ Sch/k, and denote by Sch/S the category of k-schemes over S. There is a long
list of Grothendieck topologies that can endow Sch/S, see for example [Sta, Chapter :
Topologies on Schemes].
For example the Zariski topology on Sch/S has as covering families the collections
fα
{Uα GGA X}α where each morphism fα is an open immersion and
S
fα (Uα ) = X.
The étale topology also defines a Grothendieck topology on Sch/S, where a collection
fα
of morphisms {Uα GGA X}α is defined to be a covering family if and only if each
S
morphism fα is étale and again fα (Uα ) = X.
23
CHAPTER 1. PREREQUISITES
Motivic homotopy theory is defined in terms of a topology strictly in between the Zariski
and the étale, the Nisnevich topology, which is defined and motivated in Section 3.4.
There are other conditions that can be imposed on the morphisms such as smooth,
syntomic, . . . , where the result generates a Grothendieck topology thanks to the fact
that all these properties are preserved by pullbacks and by composition, see for example
Proposition 6.8.3 in [Gro65].
(4) As in the case of topological spaces, all the topologies on the small site Sch/S extend
to topologies on the big site Sch/k of all k-schemes. Moreover, there are many other
useful subcategories of Sch/k that admit Grothendieck topologies, for example the subcategories Sm/k of smooth k-schemes, the subcategory of schemes of finite type, . . . .
Let C be a small category. Recall that the category of functors [C op , Set], sometimes
op
denoted by SetC or by Pre(C), is called the category of presheaves (of sets) on C, and a
functor F : C op GGA Set is called a presheaf (of sets). The smallness condition on C ensures
op
that the category of functors SetC is a (locally small) category.
1.3.2
Sheaves on Grothendieck sites
On a small Grothendieck site, the notion of coverings gives a sense to a local to global
iα
principle, as it is the case in the prototype category top(X). Let {Uα ,GGA U }α be an open
cover of U in top(X) and let
F : top(X)op GGA Set
be a presheaf (on X). A collection {fα }α of elements fα ∈ F (Uα ) is called compatible if,
with the notations of the two commutative squares
Uα ∩ Uβ
iαβ
Uα
iβα
Uβ
iα
iβ
F (iα )
F (U )
=⇒
F (iβ )
U
F (Uβ )
F (iαβ )
F (iβα )
for any α 6= β, the equality is satisfied
F (iαβ )(fα ) = F (iβα )(fβ )
∈ F (Uα ∩ Uβ ).
A presheaf F is called a sheaf if
• for any open cover {Uα ,GGA U }α , and
• for any compatible collection {fα },
there exists a unique element f ∈ F (U ) such that
F (iα )(f ) = fα
24
F (Uα )
for all α.
F (Uα ∩ Uβ ),
CHAPTER 1. PREREQUISITES
Example 1.4 (Sheaf on top(X)). Let X ⊆ Rn be a real manifold and consider
F : top(X)op GGA Set : U G
[ GA F (U ) = C 0 (U, R)
i
the presheaf of R-valued continuous functions. The functor F sends an embedding U ,GGA V
to the precomposition i∗ : F (V ) GGA F (U ), which is in fact the restriction to U
i∗ : C 0 (V, R) GGA C 0 (U, R) : f G
[ GA f
U.
iα
Let now {Uα ,GGA U }α be an open cover of U . A collection of elements {fα } where
fα ∈ F (Uα ) for all α, i.e., fα : Uα GGA R is compatible if
fα
Uα ∩Uβ
= fβ
Uα ∩Uβ
for any α, β such that Uα ∩ Uβ 6= ∅. Therefore, if {fα } is a compatible collection, then there
is an element f ∈ F (U ) such that each restriction to Uα is fα . Moreover, this function is
necessarily given by
f : U GGA R : u G
[ GA fα (u)
for any α such that u ∈ Uα ,
which is a well-defined function by the compatibility of the collection of functions {fα }.
Since this condition is verified for any collection of compatible elements and for any open
cover, it follows that F = C 0 (−, R) is a sheaf.
With the notation of the the above two commutative squares, the morphisms F (iβα ) and
F (iαβ ) give, for a fixed α, two morphisms of F (Uα ) into the term F (Uα ∩ Uβ ) = F (Uβ ∩ Uα )
and this induces the diagram
Q
Y
F (iαβ )
Y
F (Uγ )
Q
γ
F (iβα )
F (Uα ∩ Uβ ).
α,β
A collection of elements {fγ }γ is an element in the product F (Uγ ), and the fact that it is
Q
compatible is the same as saying that it gets equalized by the two morphisms F (iαβ ) and
Q
F (iβα ). Moreover, the fact that there exists an element f ∈ F (U ) such that F (iα )(f ) = fα
for all α means that the two composites
Q
Q
F (U )
F (iγ )
Q
Y
F (iαβ )
Y
F (Uγ )
Q
γ
F (iβα )
F (Uα ∩ Uβ ).
α,β
are the same, and the fact that such an f is unique means that F (U ) is in fact the equalizer
iα
of this diagram. Therefore, a presheaf F is a sheaf if and only if, for any open cover {Uα ,GGA
U }α in top(X), the induced diagram is an equalizer. Moreover, since the intersection Uα ∩Uβ
can be interpreted as the pullback Uα ×U Uβ , this categorifies the notion of a sheaf.
Definition (Sheaves on a Grothendieck site). Let C be a small Grothendieck site. A presheaf
F : C op GGA Set is called a sheaf if for any open covering {Uα GGA X}α in the site C, the
induced diagram
25
CHAPTER 1. PREREQUISITES
Q
F (X)
Q
F (iγ )
Y
F (iαβ )
Y
F (Uα )
Q
α
F (iβα )
F (Uα ×X Uβ )
α,β
is an equalizer in Set. In the category of presheaves, the full subcategory of sheaves will be
denoted by Sh(C) ⊆ Pre(C).
If we define the indiscrete topology to have as covering families only the isomorphisms
∼
=
{X GGA Y } ⊆ Mor(C), the sheaf condition is satisfied by any presheaf, and thus the category of sheaves for this topology is the category of all the presheaves Pre(C). In particular,
we also study the category of presheaves by only studying the sheaves on a Grothendieck
site.
In the functor category of presheaves [C op , Set], the limits and colimits are computed
object-wise, and therefore Pre(C) is a complete and cocomplete category, since Set is.
An important question about the subcategory of sheaves Sh(C) ⊆ Pre(C) is to see how
limits and colimits are computed, and to see if it is either complete or cocomplete. Since
Sh(C) is also a functor category (as Pre(C)), the right notion of limit and colimit is the
object-wise one, i.e., the limit and colimit computed in the larger category of presheaves.
Consider X : J GGA Pre(C), a J -diagram such that X(j) is in fact a sheaf for each j ∈ J ,
that is, they satisfy the condition
Q
X(j)(U )
X(j)(iγ )
Q
Y
X(j)(iαβ )
Y
X(j)(Uα )
Q
α
X(j)(iβα )
X(j)(Uα ×X Uβ )
α,β
for each covering in the site {Uα GGA U }α . The limit presheaf limj∈J X(j) of this diagram
is in fact again a sheaf, since the fact that limits commutes with limits (see for example
Section IX.8 in [Mac71]) gives that the diagram
Q
(limj∈J X(j))(U )
X(j)(iγ ) Y
α
Q
(limj∈J X(j))(Uα ) Q
X(j)(iαβ ) Y
(limj∈J X(j))(Uα ×U Uβ )
X(j)(iβα ) α,β
is isomorphic to the diagram
Q
limj∈J
X(j)(U )
X(j)(iγ )
Q
Y
X(j)(iαβ )
X(j)(Uα )
Q
α
!
Y
X(j)(iβα )
X(j)(Uα ×U Uβ ) ,
α,β
which is an equalizer since object-wise it is. The category Sh(C) of sheaves is therefore
complete and limits are computed as in the category of presheaves, i.e., object-wise. In
particular, the inclusion functor i : Sh(C) ,GGA Pre(C) preserves limits, and may therefore
admit a left adjoint. In fact, if the inclusion functor admits a left adjoint, this left adjoint
will help compute the colimits in Sh(C), by first computing it in Pre(C) and the pushing it
back to Sh(C) since left adjoints preserve colimits.
26
CHAPTER 1. PREREQUISITES
Theorem 1.3.1 (Sheafification). Let C be a small Grothendieck site. The inclusion of
sheaves into presheaves i : Sh(C) ,GGA Pre(C) admits a left adjoint
GGA
a : Pre(C) DG
G ⊥ G Sh(C) : i,
called the sheafification functor.
Idea of the proof. A proof with more details can be found in Chapter 2 of [Art62]. For any
object U ∈ C, define a category JU with
• Objects : coverings {Uα GGA U }α of U ;
f
• Morphisms : an arrow {Uα GGA U }α∈A GGA {Vβ GGA U }β∈B is a function A GGA B
fα
with morphisms Uα GGA Vf (α) ∈ C such that all triangles commute
Uα
fα
Vf (α)
U.
For any U ∈ C, a presheaf F induces a functor


Y
Y
GGA
F (Uα ×U Uβ ) ,
[ GA eq  F (Uα ) GGA
FU : JUop GGA Set : {Uα GGA U }α G
α
α,β
φ
where the equalizer is F (U ) if F is a sheaf. Moreover, any morphism V GGA U ∈ C in the
site also induces a functor
J(φ) : JU GGA JV : {Uα GGA U } G
[ GA {Uα ×U V GGA V }.
The following square
Uα ×U V
Uα ×U Uβ ×U V
Uα
Uα ×U Uβ
commutes, and the upper right corner can be replaced by Uα ×U V ×V Uβ ×U V . There
are therefore induced natural transformations FU =⇒ FV ◦ J(φ) and thus a morphism
colim FU GGA colim FV . This defines a functor
(−)+ : Pre(C) GGA Pre(C) : F G
[ GA F +
which is defined by F + (U ) := colimJU FU . This is where the magic happens, because in
turns out that F ++ is in fact a sheaf for any presheaf F . This follows from the following
result.
27
CHAPTER 1. PREREQUISITES
Fact (Lemma 2.1.2 in [Art62]). Let F ∈ Pre(C) be a presheaf.
• If for all coverings {Uα GGA U }α the natural map
F (U ) ,GGA
Y
F (Uα )
α
is an injection, then F + is a sheaf.
• The presheaf F + satisfies the above condition.
To show that the sheafification is left adjoint to the inclusion functor, observe first that
for every covering {Uα GGA U }α there is a canonical map


Y
Y
GGA
F (U ) GGA FU ({Uα GGA U }α ) = eq  F (Uα ) GGA
F (Uα ×U Uβ )
α
α,β
that is induced by F (U ) GGA F (Uα ). Moreover, since JU is connected, the colimit of the
constant diagram
JUop GGA Set : {Uα GGA U }α GGA F (U )
Q
is canonically isomorphic to F (U ). This induces a map between the colimits, and therefore
a map of presheaves F GGA F + , which is an isomorphism if F is already a sheaf. Therefore,
any morphism of presheaves F GGA G, where G is a sheaf gives
F+
F
G∼
= G+ ,
that commutes by naturality, and thus any morphism F GGA G where G is a sheaf factorizes
trough F + . In other words, if we denote by Pre(C)+ the full subcategory of Pre(C) we restrict
the objects to the ones of the form F + for any presheaf F , the functor (−)+ is left adjoint
to the inclusion
GGA
(−)+ : Pre(C) DG
G ⊥ G Pre(C)+ : i.
Moreover, by applying twice the functor (−)+ , since F ++ is a sheaf for any presheaf F , i.e.,
Pre(C)++ ∼
= Sh(C), we get the desired adjunction
GGA
(−)++ : Pre(C) DG
G ⊥ G Sh(C) : i.
The sheafification functor will usually be denoted either by (−)++ , or by a, or by L2 .
Given X : J GGA Pre(C) a J -diagram of presheaves such that X(j) is in fact a sheaf
for each j ∈ J , its colimit colim X(j) ∈ Pre(C) does not necessarily give a sheaf. This
is due to the fact that colimits do not need to commute with limits (in particular, the
equalizer that defines sheaves). As pointed out before, the right notion of a colimit in the
functor category Sh(C) is the object-wise one, i.e., the colimit computed in the category of
28
CHAPTER 1. PREREQUISITES
presheaves. Moreover, since the sheafification is left adjoint to the inclusion and in particular
preserves colimits, we can safely push the limit back in the category of sheaves
++
colim X(j)
∼ colim X(j)++ ∼
=
= colim X(j) ∈ Sh(C),
where the first two colimits are computed in the category of presheaves.
As a summary, the limit of a diagram of sheaves computed in the category of presheaves
is again a sheaf, since the inclusion i : Sh(C) ,GGA Pre(C) preserves limits. Therefore, limits
of sheaves seen as presheaves are already sheaves and are simply computed object-wise.
This property does not hold for colimits, and colimits of sheaves seen as presheaves are not
necessarily sheaves. Using the adjunction
GGA
(−)++ : Pre(C) DG
G ⊥ G Sh(C) : i,
we compute object-wise the colimit in the larger category [C op , Set] of presheaves, and push
the limit back in Sh(C) by using the right adjointness of the sheafification functor.
Given a complete and cocomplete category M, there is a more general notion of presheaves
and sheaves on C with values in M, which are functors in the category
[C op , M].
If M is complete and cocomplete, the same results as for presheaves of sets hold. In
particular :
• For any small category C, the category of presheaves [C op , M] is complete and cocomplete and limits and colimits are computed object-wise.
• If in addition C is a Grothendieck site, there is a similar notion of a sheaf. A presheaf
F : C op GGA M is a sheaf if and only if, for every covering {Uα GGA U }α in the topology
of C, the induced diagram
Q
F (X)
F (iγ )
Q
Y
F (iαβ )
Y
F (Uα )
Q
α
F (iβα )
F (Uα ×X Uβ )
α,β
is an equalizer in M. Denote the subcategory of sheaves by Sh(M) ⊆ Pre(M).
• The inclusion functor i : Sh(M) ,GGA Pre(M) is part of an adjunction
GGA
L2 : Pre(M) DG
G ⊥ G Sh(M) : i.
• Limits of sheaves are computed object-wise and colimits of sheaves are given by the
sheafification of the colimit of the underlying presheaves (sheafification of the objectwise colimit).
As a general idea, all the structure of the category of values M is (object-wise) inherited
by the category of presheaves [C op , M], and some of it by the subcategory of sheaves Sh(M)
by using the adjunction
GGA
L2 : Pre(M) DG
G ⊥ G Sh(M) : i.
29
CHAPTER 1. PREREQUISITES
For example, the abelian structure of an abelian category M is (object-wise) inherited
by the category presheaves Pre(M), and after sheafification by the subcategory of sheaves
Sh(M). Moreover, if M has enough injectives, then so does Pre(M), but not necessarily
Sh(M).
In this project, we will be interested in presheaves with values in simplicial sets, that
is, categories of functors [C op , sSet]. The homotopy theory of simplicial sets will give a
homotopy theory on [C op , sSet] and most of the work in this project is to find such a
suitable homotopy theory.
1.4
Simplicial and Cosimplicial Objects
Given a small category C, the category [C op , sSet] of presheaves on C with values in simplicial
sets sSet is called the category of simplicial presheaves (on C). A functor in this category
F : C op GGA sSet = [∆op , Set]
can also be seen as a bifunctor on C op and ∆op
F : C op × ∆op GGA Set,
i.e., a presheaf on the product C × ∆, or as a functor
F : ∆op GGA [C op , Set],
which is a simplicial object in the category of presheaves [C op , Set]. This section treats with
the very basic definition of simplicial and cosimplicial objects in arbitrary categories, which
are very useful in homotopy theory. As we will see, the category sSet of simplicial sets
plays a very important role in homotopy theory, in some sense, it plays a similar role than
does the category Set of sets in the theory of (locally small) categories. In fact, by adding
a simplicial dimension to the objects in M, a homotopy theory of these objects naturally
arises.
1.4.1
(Co)simplicial objects and (co)skeletons
Definition (Simplicial and cosimplicial objects). A simplicial object in a category M is a
functor
∆op GGA M,
and a cosimplicial object in a category M is a functor
∆ GGA M.
For example, a simplicial object in the category Set of sets is a simplicial set. Alternatively, as in the category of simplicial sets, a simplicial object in a category M can be given
as a sequence of objects M0 , M1 , M2 , . . . with face maps di : Mn GGA Mn−1 and degeneracy
maps sj : Mn GGA Mn+1 , satisfying the simplicial identities.
In a general category M, say complete, an example of a simplicial object is the Cech
complex of an object X
30
CHAPTER 1. PREREQUISITES
···
X ×X ×X
X ×X
X,
where we did not draw the degeneracies for visual reasons. More generally, any morphism
X GGA Y ∈ M defines a Cech complex
···
X ×Y X ×Y X
X ×Y X
X,
which is a simplicial object in M.
Denote the n-truncated simplicial category
0
1
..
.
···
2
n,
by ∆n . This is the full subcategory of the simplicial category ∆, with objects 0, 1, . . . , n.
op induces by precomposition a functor
The inclusion functor ∆op
n ,GGA ∆
i∗ = trn : M∆
op
op
GGGGA M∆n ,
called the n-truncation. As its name indicates, this functor sends a simplicial object M• to
its n-truncated part M0 , M1 , . . . , Mn with the same faces and degeneracies.
Let’s now assume that M is complete and cocomplete. Since ∆op is small, left and right
Kan extensions give left and right adjoints to the precomposition by i functor
i∗ = trn : M∆
op
op
GGGGA M∆n .
The left adjoint given by left Kan extension is called the n-skeleton
∆op
fn : M∆n GGA
sk
: trn ,
DG
G ⊥G M
op
and the right adjoint, defined by right Kan extension is called the n-coskeleton
trn : M∆
op
∆op
gn
n
GGA
: cosk
DG
G ⊥G M
op is full subcategory, the unit
In addition, since ∆op
n ⊆∆
∼
=
op
M• GGA trn ◦ skn M•
and the counit
∼
=
∈ M∆n
trn ◦ coskn M• GGA M•
op
∈ M∆n
are isomorphims (see Corollary 4 in Section X.3 in [Mac71]). By abuse of language, the two
composites
fn ◦ trn : M∆
skn = sk
op
GGA M∆
op
and
g n ◦ trn : M∆
coskn = cosk
op
op
GGA M∆ ,
are also called the n-skeleton and the n-coskeleton, and we will be careful to specify which
functor we consider, when needed. Therefore, for an n-truncated simplicial object X• its
skeleton and coskeleton are defined by
fn X•
sk
and
31
g n X• ,
cosk
CHAPTER 1. PREREQUISITES
while for a simplicial object X• , its skeleton and coskeleton are defined as the skeleton and
coskeleton of its n-truncated part. In particular, the n-skeleton and n-coskeleton do not
see the information contained outside of the n-truncation. Moreover, for a simplicial object
op
op
X• ∈ M∆ and an n-truncated simplicial object Y• ∈ M∆n , the adjunctions skn a trn a
coskn give two natural bjections
op
op
op
op
M∆ (skn (Y• ), X• ) ∼
= M∆n (Y• , trn (X• )) and M∆n (trn (X• ), Y• ) ∼
= M∆ (X• , coskn (Y• )).
op
In particular, given an n-truncated simplicial object Y• ∈ M∆n there are two universal
ways in how to extend it to a simplicial object :
• the skeleton skn Y• is the way such that any map from it to a simplicial object is
determined by what it does on the n-truncation;
• the coskeleton coskn Y• is the way such that any map into it from a simplicial object is
determined by what it does on the n-truncation.
1.4.2
Augmented simplicial objects
An augmented simplicial object is defined as a simplicial object with an extra object M−1
in the −1th dimension with a unique morphism M0 GGA M−1 . As a concrete example, an
augmented simplicial set is a simplicial set K• with an extra set K−1 of −1-simplices such
that each 0-simplex is associated to a unique −1-simplex. This extra set K−1 could for
example represent a set of colours, and the face map d : K0 GGA K−1 associates a colour to
each vertex.
Consider the augmented simplex category ∆+ that is ∆op with an initial object −1, i.e.,
it is defined as
−1
0
1
2
..
.
··· ,
where all possible compositions −1 GGA m give the same morphism.
Definition (Augmented simplicial object). An augmented simplicial object in M is a functor
M• : ∆op
+ GGA M.
As for simplicial objects, an augmented simplicial object in M can be seen as a sequence
of objects M−1 , M0 , M1 , . . . ∈ M with faces and degeneracies that satisfy some simplicial
identities.
A third and more useful way to interpret an augmented simplicial object is the following.
Let N be an object in M, and write rN for the constant simplicial object with rNn = N
for every n ≥ 0 and with only identity maps as faces and degeneracies. For any simplicial
object M• , a morphism of simplicial objects M• GGA rN (that is, a natural transformation)
is the same as a morphism Mm GGA N ∈ M for any fixed m, since rN has only identities as
structure maps. In particular, such a morphism M• GGA rN defines an augmentation map
M0 GGA N and this is equivalent to an augmented simplicial object M−1 = N, M0 , M1 , . . ..
32
CHAPTER 1. PREREQUISITES
In many cases, augmented simplicial objects are augmented by a morphism arising from a
colimit. Indeed, a simplicial object M• : ∆op GGA M can be interpreted as a ∆op -diagram in
M and the morphisms Mn GGA colim M• define an augmentation, i.e., M• GGA r colim M•
is an augmented simplicial object. Similarly, the natural morphism M• GGA hocolim M•
is also an augmented simplicial object. The language of augmented simplicial objects gives
a formalism in which a morphism M• GGA rN is naturally seen as an object, and more
op
precisely, an object of the category of functors M∆+ .
The notion of a morphism between augmented simplicial objects is given by natural
transformations of functors, and thus augmented simplicial objects form a category. All constructions with simplicial objects can be extended to augmented simplicial objects. In parop
ticular, the embeddings ∆op
n ,GGA ∆+ also give n-truncations, n-skeletons and n-coskeletons
by Kan extensions, i.e., there are adjunctions
∆
fn : M∆n GGA
sk
DG
G ⊥ G M + : trn
op
op
and
op
∆op
g n.
n
GGA
trn : M∆+ DG
: cosk
G ⊥G M
We will later use this language to see hypercovers as objects themselves, rather than a
morphism from a simplicial object to a constant simplicial object.
33
2. Additional Structures on Model
Categories
The language of model categories, introduced by Quillen, is an efficient machinery for doing
homotopy theory. In fact, a model category is equipped with more than enough structure
for homotopy theory, that is, it contains more data than only a class of weak equivalences.
The extra data turns out to be very useful, because it gives a presentation of its homotopy
category, which, at first sight, only looks like a localization. However, the axioms for a model
category are not too restrictive, since many categories admits (at least) such a model. For
example, the category Top of topological spaces, the category sSet of simplicial sets, the
category ChainR of chain complexes of R-modules, categories of diagrams with target a
model category, . . . , all admit model structures. There are many written introductions to
model categories, the handbook of Dwyer and Spalinski [DS95], the monograph of Hovey
[Hov99] or the one of Hirschhorn [Hir03] are all good references.
In this first section we present the definition of a model category that will be used
throughout this report. We will give all the necessary definitions as well as their first
properties in order to understand and be able to use model categories. The crucial notions
of a cofibrantly generated model category, as well as a cellular and a combinatorial model
category are explored.
The second section studies simplicial model categories, which are model categories enriched over sSet, in which the simplicial mapping space between any two objects has a
homotopy theoretic content.
In the third section we develop the notion of localization of model categories, as explained
by Hirschhorn in [Hir03]. This notion is crucial for the next chapter, where it will be used
more than once.
2.1
2.1.1
Model Categories
A few categorical prerequisites
Let C be a category. We need first introduce a few definitions that will be used in the
category of arrows Arr(C) = C 2 .
Definition (Retract). An object A ∈ Ob(C) is said to be a retract of B ∈ Ob(C) if there
id
exists a factorization of A GGA A through B
34
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
id
A
B
A.
p
i
A morphism A GGA B is itself said to be a retract, if there is a retraction B GGA A
id
A
i
B
p
A.
Moreover, a class of objects W ⊆ Ob(C) is said to be closed by retracts or closed under
retracts, if given any object B ∈ W and any retract A ∈ C of B, then A is also in W .
i
If A GGA B is a retract, it immediately follows that A is a retract of B. These terms
come from what happens in topology, where an inclusion A ,GGA X is a retract if there
f
exists a retraction X GGA
A A. Similarly, a morphism A GGA B is said to be a retract of
g
X GGA Y if it is so in the category of arrows Arr(C), i.e., if there exists a commutative
diagram in C
id
A
X
A
g
f
B
f
Y
B.
id
If the objects of C have an underlying set, and if a morphism A GGA B ∈ Arr(C) has an
underlying set-theoretic function, if A is a retract of B
id
A
i
B
p
A,
then i is necessarily injective and p is necessarily surjective. For example, in the category
Set of sets, the retraction are exactly the surjective maps1 . In the category R Mod of
R-modules, M ,GGA N is a retract if and only if there is another R-module L such that
M∼
= N ⊕ L.
There are several useful immediate results2 such as the fact that the retract of a monomorphism, epimorphism, isomorphism, is again respectively a monomorphism, epimorphism or
1
2
This statement is equivalent to the axiom of choice.
See for example Section 7 in [AHS06].
35
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
isomorphism; retractions are closed under composition; any retraction is an epimorphism;
any functor preserves retract(ion)s; any fully faithful functor reflects retract(ion)s, . . . . Another property that arrows can have in the category Arr(C) is a left/right lifting property
with respect to another arrow.
f
Definition (Left/Right lifting property). We say that an arrow A GGA B has the left lifting
g
property with respect to X GGA Y , or equivalently, that g has the right lifting property with
respect to f , if every commutative square in C
A
X
g
f
B
Y
admits a diagonal filler that makes the two triangles commute.
Note that isomorphisms have both the left and right lifting property with respect to
f
any other morphism. Moreover, if A GGA B has the left (resp. right) lifting property with
g
respect to X GGA Y , then any retract of f has the left (resp. right) lifting property with
respect to g. The last property that we need in order to define model categories, is the
2-out-of-3 property of a class of arrows.
Definition (2-out-of-3 property). A class of morphisms W ⊆ Arr(C) is said to have the
2-out-of-3 property in C, if for any two composable morphisms f and g, if two out of the
three morphisms f, g and f ◦ g are in W , then so is the third.
In particular, such a class W is closed under composition. The standard example of
such a class of morphisms is the class of isomorphisms in any category, because they have
an inverse. In fact, the 2-out-of-3 property is meant to be a weakening of the property of
being an isomorphism. The class of morphisms that will have this property will behave
similarly to isomorphisms, without necessarily having an inverse morphism. We have now
all the ingredients to define a model category.
2.1.2
The definition of a model category and examples
Definition (Model category). A model category is a category M with three classes of morphisms C , F , W ⊆ Mor(M) called the cofibrations, the fibrations and the weak equivalences
satisfying the following axioms
(CM1) The underlying category M is complete and cocomplete;
(CM2) The class of weak equivalences W has the 2-out-of-3-property;
(CM3) The three classes C , F and W are closed under retracts;
(CM4) Any commutative square
36
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
A
X
H
i
p
B
Y
where i ∈ C is a cofibration, p ∈ F is a fibration, and either i or p is a weak equivalence,
admits a lifting H;
f
(CM5) Any morphism A GGA B admits two functorial factorizations
f
A
B
p
i
f
A
q
j
C
B,
C
where i ∈ C ∩ W is a cofibration and a weak equivalence and p ∈ F is a fibration, and
where j ∈ C is a cofibrations and q ∈ F ∩ W is a fibration and a weak equivalence.
In addition, a morphism in C ∩ W is called an acyclic cofibration and a morphism in
F ∩W is called an acyclic fibration. Moreover, as already done in the definition, a cofibration
will often be decorated as X GA Y , a fibration as X GGA
A Y and a weak equivalence as
∼
X GGA Y . Since there is usually no confusion, we will use the symbol M both for the
underlying category and for the model category, i.e., the category M with the extra data
of cofibrations C , fibrations F and weak equivalences W . When we need to be precise, we
will refer to the model category as the quadruple (M, C , F , W ).
Note that since the category is complete and cocomplete, there is a initial object, denoted
by ∅, and a terminal object, denoted by ∗. An object X is called cofibrant if the unique
morphism ∅ GA X is a cofibration, fibrant if the unique morphism X GGA
A ∗ is a fibration,
and bifibrant if it is both cofibrant and fibrant. The theory of model categories would
be much easier if all objects would be bifibrant. In fact, these objects allow extensions
and liftings of morphisms, in a similar manner than projective modules allow liftings of
epimorphisms and injective modules allow extensions of monomorphisms. For example, if
∼
A is cofibrant and X GGA
A Y is an acyclic fibration, axiom (M4) gives a diagonal filler
!
∅
!
H
X
∼
A
Y,
37
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
that is exactly the same as a lifting of the morphism A GGA Y
X
H
∼
A
Y.
Dually, if X is fibrant, any morphism A GGA X admits an extension along any acyclic
∼
cofibration A GA B
A
X.
∼
H
B
These axioms of a model category are slightly stronger than the original definition of
Quillen in [Qui67]. First, Quillen only asks for finite limits and finite colimits to exist in
the underlying category, instead of requiring all (small) limits and colimits. However, most
categories that admit interesting model structures are complete and cocomplete3 . Second,
Quillen did not ask for the functoriality of the two factorizations. However, it is rather
hard to come up with a model category that does not admit functorial factorizations, see
for example the Remark 4.10 in [Isa04].
To explain what the functoriality is, the best approach is to see the category of morphisms
of M as the functor category M2 . There is a composition functor
f
g
g ◦f
M3 GGA M2 : A → B → C G
[ GA A → C.
Observe that choosing a factorization for each map f ∈ Mor(M) = M2 by, say an acyclic
cofibration if followed by a fibration pf , such as
f
Xf
if
Yf
pf
∼
Af ,
is the same as giving an association
if
f
pf
M2 GGA M3 : Xf → Yf G
[ GA Xf → Af → Yf
which is a section of the composition functor and where if ∈ C ∩ W and pf ∈ F . Asking
for a functorial factorization is asking that this is in fact a functor (and similarly for the
3
The category of bounded chain complexes is an example of a category that is only finitely complete and
cocomplete.
38
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
other factorization as a cofibration followed by an acyclic fibration). This means that given
a morphism (g1 , g2 ) : f GGA f 0 in M2
f
Xf
Yf
g1
g2
Xf0
f0
Yf0 ,
the choice of the terms Af and A0f is functorial, and there are commutative squares
Xf
if
Af
g1
Xf0
pf
Yf
g3
g2
A0f
i0f
Yf0 .
p0f
This interpretation of a functorial factorization is taken from the excellent article [Gar09],
where this is explained and motivated in much greater depth. Moreover, there is an equivalent, more compact, definition of a model category defined with the help of weak factorization
systems.
Definition (Weak factorization system). A weak factorization system4 on a category C is a
pair of classes of maps (L, R) that are closed under retracts, and satisfying the two axioms
that
• each morphism f ∈ Mor(C) factorizes as a map from L followed by a map from R;
• every pair of morphisms (l, r) ∈ L × R has the lifting property, i.e., any commutative
square in C admits a filler as in the diagram
A
l
X
∃
B
r
Y.
Equivalently to our definition, a model structure on a bicomplete category M is a class W
of weak equivalences that has the 2-out-of-3 property, together with two weak factorization
systems (C ∩ W , F ) and (C , F ∩ W ). Weak factorizations systems, as well as other types
of factorization systems are treated in this same article [Gar09].
An important further observation is that in a model category, two of the three classes
C , F and W completely determine the third. Indeed, it follows from the definition that all
4
The word weak is here to remind us that this factorization is not necessarily unique.
39
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
fibrations have the right lifting property with respect to acyclic cofibrations. In fact, the
fibrations turn out to be exactly the morphisms that have the right lifting property with
f
respect to all acyclic cofibrations. Suppose that a morphism X GGA Y has the right lifting
property with respect to all acyclic cofibrations, and choose a factorization
f
X
i
Y
p
∼
Z.
Since i is an acyclic cofibration, the following square has a diagonal filler
id
X
X
∃g
f
i
Z
p
Y
and then it follows that f is a retract of the fibration p, and so is itself a fibration. A
similar argument shows that the cofibrations are exactly the morphisms that have the left
lifting property with respect to all acyclic fibrations. For the last case, if we know all
the cofibrations and all the fibrations, this argument shows that we know all the acyclic
cofibrations and all the acyclic fibrations. By the 2-out-of-3 property, a morphism X GGA Y
is a weak equivalence if and only if it can be written as a composite of an acylic cofibration
and an acyclic fibration.
The fact that two classes determines the third relies on the fact that we are given two
weak factorization systems. In a weak factorization system (L, R), one of the two classes L or
R determines the other. Moreover, the fact that the cofibrations are exactly the morphisms
that have the (left) lifting property with respect to all acyclic fibrations, implies that the
class of cofibrations are closed under composition. Similarly, the class of acyclic cofibrations,
fibrations and acyclic fibrations are also closed under composition. More precisely, C , F
and W are subcategories of Mor(M), containing all the objects as a domain or codomain
of an arrow, and containing all isomorphisms.
Given a model category M with cofibrations C , fibrations F and weak equivalences W ,
there is a canonical model category on the opposite underlying category Mop where the
cofibrations are F op , the fibrations are C op and the weak equivalences are W op .
Example 2.1 (Example of model categories).
(1) Any complete and cocomplete category M, admits two (not really interesting) model
structures. These are obtained by setting the weak equivalences and one of C or F to be
all the morphisms in Mor(M), while the last one contains exactly all the isomorphisms.
The axioms are easily checked.
40
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
(2) Given a commutative ring R with unit, the category ChainR,≥ of positive chain complexes of modules over R (positively graded differential R-modules) admits two model
structures. Both model have as weak equivalence the quasi-isomorphisms, i.e., the
maps inducing isomorphisms in homology at every level. In the projective version, the
fibrations are the degree-wise epimorphisms while the cofibrations are the degree-wise
monomorphisms with projective cokernel, and in the injective version, the cofibrations
are the degree-wise monomorphisms while the fibrations are the degree-wise epimorphisms with injective kernel. The extra condition for the cofibrations in the projective
version is required so that they have the left lifting property with respect to the acyclic
fibrations (degree-wise epimorphisms that are quasi-isomorphisms), and similarly for
the extra condition for the fibrations in the injective model. Since the chain with only
the 0 R-module is both initial and terminal, every object is cofibrant in the injective
model, and every object is fibrant in the projective model.
(3) There are two well-known model structures on the category Top of all topological spaces,
which, in a sense to be defined below, give the same homotopy theory. The first is due
to Quillen, and has as weak equivalences the weak homotopy equivalences, which are
the maps inducing isomorphisms on all homotopy groups, for any choice of basepoint.
The (Serre) fibrations are the maps satisfying the right lifting property with respect to
the inclusions Dn ∼
= Dn × {0} ,GGA Dn × I, and the cofibrations are, roughly speaking
the maps that can be built out of the natural inclusions Sn ,GGA Dn+1 . See the Section
2.4 in [Hov99] for a complete proof that these definitions give a model structure on Top.
In this model, every object is fibrant.
(4) The second model structure on Top is the Hurewicz model (or the Strom model), first
proved by Strom in [Str72]. The weak equivalences are in this model only the homotopy
equivalences, i.e., the maps having an inverse up to homotopy. There a fewer fibrations
since the fibrations are the Hurewicz fibrations, i.e., the maps having the lifting property
with respect to all inclusions X ∼
= X × {0} ,GGA X × I for every topological space X.
The cofibrations are the closed Hurewicz cofibrations. This model has the nice property
that every space is both cofibrant and fibrant.
(5) The standard model structure on the category sSet of simplicial sets, gives again the
same homotopy theory as topological spaces. As a consequence, it is sometimes more
convenient to work with simplicial sets instead of topological spaces, while interested in
properties up to homotopy. The weak equivalences are the maps X· GGA Y· that are
weak homotopy equivalences in Top after the geometric realization. The cofibrations
are exactly the monomorphisms, i.e., the maps that are injective in each degree. The
fibrations are the Kan fibrations, which are the maps satisfying the right lifting property
with respect to all horn inclusions Λkn ,GGA ∆[n]. Every object is trivially cofibrant,
and the fibrant objects are called the Kan complexes.
2.1.3
The construction of the homotopy category
A very convenient aspect of model categories is that they give a presentation of their homotopy categories. More precisely, the associated homotopy category is constructed as a
certain localization of the model category.
41
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Definition (Homotopy category). Given a model category M, its homotopy category, if
it exists, is the localization M[W −1 ] at the class of weak equivalences, usually denoted by
Ho(M).
L
The localization is more precisely a functor M GGA Ho(M), where every weak equivalence in M becomes an isomorphism in the homotopy category Ho(M), and which satisfies
F
the universal property that for any other such functor M GGA A, there is a filler, unique
up to unique isomorphism in the diagram
M
L
F
A.
∃! θ
Ho(M)
Observe that the homotopy category does only depend on the weak equivalences. Therefore, any two model structures with same weak equivalences on the same category M, will
have isomorphic homotopy categories. The extra structure of a model category, given by
the cofibrations C and the fibrations F , allows an explicit construction of the homotopy
category of any model category. We will briefly sketch this construction, and refer to Section
5 in [DS95] or Section 1.2 in [Hov99] for more details. The idea is to introduce a homotopy
equivalence relation, and to quotient the class of morphisms so that the maps that have
inverses up to homotopy will become isomorphisms. The next step is to show that if we
restrict our attention to bifibrant objects, the weak equivalences of M are exactly the maps
that have an inverse up to homotopy, and so the work is done for bifibrant objects. Finally,
we extend this construction to all objects by using a ’homotopy-invariant’ functor from the
model category to its full subcategory of bifibrant objects.
Mimicking the construction of homotopies from Top, the first ingredient is to define
some sort of cylinder object that plays the role of X × I for a fixed object X. Note that in
Top, there are two injections
i0
X∼
= X × {0} ,GGA X × I
and
i1
X∼
= X × {1} ,GGA X × I,
that play an important role in the definition of a homotopy. Moreover, the projection
∼
X × I GGA X is a weak equivalence, since I is contractible.
Definition (Cylinder object). In a model category, a cylinder object for an object X is a
factorization of the codiagonal map
X
`
X
codiag
X,
∼
Cyl X
42
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
∼
where we only ask for Cyl X GGA X to be a weak equivalence. Such a cylinder is called
`
good if X X GA Cyl X is a cofibration, and very good if in addition Cyl X GGA
A X is a
i0
i1
fibration. There are two natural composite denoted by X GGA Cyl X and by X GGA Cyl X.
Very good cylinders always exist by factoring the codiagonal map X X GGA X into
a cofibration followed by an acyclic fibration. In the Quillen model structure on Top, the
object X × I is in general only a cylinder object, while it is at least a good cylinder object
in the Hurewicz model. In the model structure on simplicial sets, the object X × ∆[1] is
also at least a good cylinder object. Such an object allows the notion of a left homotopy
between two morphisms to be defined.
`
GGA Y are left
Definition (Left homotopy). In a model category, two morphisms f, g : X GGA
homotopic, denoted by f ∼l g if there exists a filler in the diagram
i0
X
Cyl X
i1
X,
∃
g
f
Y
for some cylinder object of the source X.
For any pair of objects X, Y ∈ M, this is a reflexive and symmetric relation on M(X, Y ).
It need not be transitive, but it is if in addition X is cofibrant. There is a dual notion to
this notion of left homotopy.
Definition (Cocylinder object). In a model category, a cocylinder object or a path object
for an object X is a factorization of the diagonal map
diag
X
X × X,
∼
Cocyl X
∼
where we only ask for X GGA Cocyl X to be a weak equivalence. Such a cocylinder is called
good if Cocyl X GGA
A X × X is a fibration, and very good if in addition X GA Cocyl X
ev0
is a cofibration. There are two natural composites denoted by Cocyl X GGA X and by
ev1
Cocyl X GGA X.
Similarly very good cocylinders always exist by factoring the map X GGA X X into
an acyclic cofibration followed by a fibration. In the category of topological spaces, the
standard path object of X is the space Top(I, X) with the compact-open topology, and ev0
and ev1 are the evaluation at 0 and 1. The notion of a right homotopy is the dual definition
of a left homotopy.
Q
43
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
GGA Y are
Definition (Right homotopy). In a model category, two morphisms f, g : X GGA
right homotopic, denoted by f ∼r g if there exists a filler in the diagram
Y
ev0
Cocyl Y
ev1
Y
∃
g
f
X,
for some cocylinder object of the target Y .
Similarly, this is a reflexive and symmetric relation, which is also transitive if Y is fibrant.
Therefore, if X is cofibrant and Y is fibrant, both ∼l and ∼r are equivalence relations on
M(X, Y ). In the category of topological spaces, the exponential adjunction5
Top(X × I, Y ) ∼
= Top(X, Top(I, Y )),
and the fact that X × I is a cylinder and Top(I, Y ) is a cocylinder, implies that the notions
of left and right homotopy agree on Top. Moreover if X is cofibrant and Y is fibrant, the
two equivalence relations agree ∼l =∼r on Top(X, Y ). This is the homotopy equivalence
relations, denoted by ∼. Given two objects bifibrant X and Y , denote by [M](X, Y ) the
quotient set of morphisms M(X, Y ) /∼ . Moreover, for any three objects X,Y and Z that
are cofibrant and fibrant, the composition on the quotient set of maps is a well-defined
function
[M](X, Y ) × [M](Y, Z) GGA [M](X, Z) : ([f ], [g]) G
[ GA [g ◦ f ].
Therefore, this gives a method to construct the localization of the full subcategory of bifibrant objects, with respect to the class of homotopy equivalences between bifibrant objects.
However, the homotopy category is the localization (of the whole category) at the class
of weak equivalences. In fact, between bifibrant objects, weak equivalences and homotopy
equivalences are the same notion.
Lemma 2.1.1. Let X and Y be bifibrant objects in a model category M. A morphism
X GGA Y is a weak equivalence if and only if it is a homotopy equivalence.
Proof. This is Lemma 4.24 in [DS95].
Therefore, between bifibrant objects, the hom-set in the homotopy category may look
like [M](X, Y ). The last idea is based on the fact that any object in the model category
may be ’replaced’ by a weakly equivalent bifibrant object. This is done by applying both a
cofibrant replacement and a fibrant replacement.
Definition ((Co)fibrant replacement). Let X ∈ M be an object in a model category. A
cofibrant replacement of X is a factorization of the morphism ∅ GGA X as
5
Which applies since I is a locally-compact and Hausdorff topological space.
44
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
∅
X.
∼
QX
Similarly, a fibrant replacement of X is a factorization of X GGA ∗ as
∗.
X
∼
RX
The point is to replace X up to a weak equivalence, by a cofibrant object QX or a fibrant
object RX. Let’s fix for any object X a cofibrant replacement QX and fibrant replacement
RX. For convenience, if X is cofibrant let QX = X, and if X is fibrant let RX = X. Then
Q extends to a functor from M to the full
. subcategory of cofibrant objects where the set of
0
morphisms is the quotient set M(C, C ) ∼l . In fact, the lifting
∅
QY
∴∃
QX
∼
Y
X
is uniquely defined up to left and right homotopy equivalence. Similarly, R extends to a
functor from M to the appropriate target category. The important result is the fact that
RQ and QR are functors from M to the full subcategory of bifibrant objects, where the set
of morphisms between two bifibrant objects X and Y is the quotient set [M](X, Y ).
Theorem 2.1.2. Let M be a model category. Its homotopy category Ho(M) always exists,
and is given by
• Objects : the same as M;
• Morphisms : the quotient sets Ho(M)(X, Y ) := M(QRX, QRY ) /∼ ;
γ
where the localization functor M GGA Ho(M) is the identity on objects, and defined as
in the paragraph before the theorem on morphisms. Moreover, a morphism f is a weak
equivalence in M if and only if its image γ(f ) is an isomorphism in the homotopy category
Ho(M).
45
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Since the homotopy category is a localization of the model category at the class of weak
equivalences, Ho(M) only depends on the class of weak equivalences W . As a consequence,
different model structures on a category M may give isomorphic homotopy categories, as a
long as they share the same weak equivalences. Having different model structures giving the
same homotopy category is often useful, as we will see in Chapter 3. A first reason when
changing the model structure is useful, is for having different classes of cofibrant and fibrant
objects, since these objects are good in that they allow extending and lifting morphisms. It
is rare that there exists a model in which every object is cofibrant and fibrant, as it happens
in Top.
2.1.4
Functors between model categories
Having introduced the homotopy category associated to a model category M, a first question
that arises is to ask when a functor F : M GGA N induces a functor from the homotopy
category Ho(M) GGA N . In some sense, this is equivalent to asking when a functor F is
compatible with the internal homotopy relation of M. Note that this does not depend on
whether or not N is a model category.
Definition (Left and right derived functor). Let M be a model category, N any category
and F : M GGA N a functor. A left derived functor of F is a functor LF : Ho(M) GGA N
with a natural transformation σ : LF ◦ γ =⇒ F , that is universal among such pairs. That
is, the triangle
F
γ
N
=⇒
M
LF
Ho(M)
commute up to the natural transformation σ, and for any other such pair (LF 0 , σ 0 ), there is
α
a unique natural transformation LF 0 =⇒ LF such that σ ◦ (α ? idγ ) = σ 0 .
Dually, a right derived functor of F is a functor RF : Ho(M) GGA N with a natural
transformation σ : F =⇒ RF ◦ γ, that is universal among such pairs. That is, the triangle
F
M
=⇒
γ
N
RF
Ho(M)
commute up to the natural transformation σ, and for any other such pair (RF 0 , σ 0 ), there
α
is a unique natural transformation RF =⇒ RF 0 such that (α ? idγ ) ◦ σ = σ 0 .
46
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
If a functor F from a model category M sends all weak equivalences to isomorphisms,
the universal property of the localized category Ho(M) says that there is a unique filler
M
F
N,
γ
LF = RF
Ho(M)
and the triangle strictly commutes (not only up to a natural transformation).
A weaker, and very often used, condition for which a functor F : M GGA N admits a
left derived functor is when F sends acyclic cofibrations between cofibrant objects to isomorphisms. Indeed, if F satisfies this property a short argument shows that F sends all
weak equivalences between cofibrant objects to isomorphisms (this is Ken Brown’s Lemma).
In this case, a possible derived functor is the composite LF = F ◦ Q for some cofibrant replacement Q. Indeed, any change of cofibrant replacement Q is only seen up to isomorphism
in N , thanks to the fact that F sends all weak equivalences between cofibrant objects to
isomorphisms in N . A dual condition holds for a right derived functor of F .
If in addition N is also endowed with a model structure, we can now be interested in when
F : M GGA N induces a functor between the homotopy categories. A similar condition as
in the previous paragraph may now be weakened, because we do not need a ’complete’ lift
Ho(M) GGA N , but only a lift in the homotopy category Ho(M) GGA Ho(N ).
Definition (Total left/right derived functor). A total left derived functor of F : M GGA N
between model categories, is a functor between the homotopy categories LF : Ho(M) GGA
Ho(N ) that is left derived functor of the composite
M GGA N GGA Ho(N ).
Similarly, a total right derived functor of F is RF : Ho(M) GGA Ho(N ) that is a right
derived functor of the same composite.
However, we are often interested in functor with more structure than just inducing a
functor on (from) the homotopy category. In order to be able to compare two homotopy
categories M and N , we would like an adjunction between them.
GGA
Definition (Quillen adjunction). An adjunction F : M DG
G ⊥ G N : G between two model
categories is called a Quillen adjunction or a Quillen pair if in addition
• either the left adjoint F sends cofibrations to cofibrations and acyclic cofibrations to
acyclic cofibrations;
• or the right adjoint G sends fibrations to fibrations and acyclic fibrations to acyclic
fibrations.
The functor F is called a left Quillen functor and G a right Quillen functor.
47
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Using the characterization of cofibrations and fibrations, the two conditions are clearly
equivalent. More precisely, the fact that F preserves cofibrations is the same as that G
preserves acyclic fibrations, and the fact that F preserves acyclic cofibrations is the same as
that G preserves fibrations. Therefore, it is also equivalent to asking one of the two mixed
conditions
• F sends cofibrations to cofibrations and G sends fibrations to fibrations;
• F sends acyclic cofibrations to acyclic cofibrations and G sends acyclic fibrations to
acyclic fibrations.
By the preceding paragraph, one of these conditions is enough to ensure that F admits a
total left derived functor LF : Ho(M) GGA Ho(N ) and that G admits a total right derived
functor RG : Ho(N ) GGA Ho(M). More importantly, the two induced functors form an
adjunction of the homotopy categories.
GGA
Proposition 2.1.3. Let F : M DG
G ⊥ G N : G be a Quillen adjunction. Then, the total left
derived functor of F and the total right derived functor of G form an adjunction on the
homotopy categories
GGA
LF : Ho(M) DG
G ⊥ G Ho(N ) : RG.
Proof. See Lemma 1.3.10 in [Hov99].
Some Quillen adjunctions not only induce an adjunction between the homotopy categories, but an equivalence of the homotopy categories. In fact, the characterization of these
special Quillen adjunctions is easy to describe.
GGA
Definition (Quillen equivalence). A Quillen adjunction F : M DG
G ⊥ G N : G is called a
Quillen equivalence, if for all cofibrant objects M ∈ M and fibrant objects N ∈ N , the
natural bijection
N (F M, N ) ∼
= M(M, GN )
restricts to a natural bijection between the sets of weak equivalences
∼ w. e. (M(M, GN )) .
w. e. (N (F M, N )) =
∼
That is to say that a morphism F M GGA N ∈ N is a weak equivalence if and only if its
∼
adjoint morphism M GGA GN ∈ M is a weak equivalence. This turns out to be a necessary
and sufficient condition for having an induced equivalence of the homotopy categories.
GGA
Theorem 2.1.4. Let F : M DG
G ⊥ G N : G be a Quillen adjunction. This is a Quillen equivalence if and only if the induced adjunction on the homotopy categories
GGA
LF : Ho(M) DG
G ⊥ G Ho(N ) : RG
is an equivalence of categories.
Proof. The proof can be found as Proposition 1.3.13 in [Hov99].
48
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Consider the realization-singular adjunction between topological spaces and simplicial
sets
GGA
Re : sSet DG
G ⊥ G Top : Sing.
We will see later that this is a Quillen pair6 . Moreover, both the unit and the counit
∼
X• GGA Sing ◦ ReX•
and
∼
Re ◦ SingX GGA X
are weak equivalences. Since they become isomorphisms at the level of the homotopy categories, the realization-singular pair is a Quillen equivalence.
2.1.5
Cofibrantely generated model categories and the small object argument
Given 3 classes of maps in a bicomplete category M, that are closed by retracts and by
composition and one of them having the 2-out-of-3-property, it is never easy to check that
this corresponds to a model structure on M. In this situation, the axioms to be checked are
the lifting property and the existence of the two (functorial) factorizations, that is, showing
the existence of two weak factorizations systems. Maybe we should emphasize the fact that
the class of weak equivalences is the most important of the three classes. Indeed, it is clear
that the homotopy category, which is the localization M[W −1 ], only depends on the class
of the weak equivalences. Therefore, the class of weak equivalences should be the first class
to be determined, in order to endow a category with a model structure. Afterwards, there is
a balance to be found between the cofibrations and the fibrations. By the lifting properties,
more cofibrations implies fewer fibrations, and more fibrations implies fewer cofibrations.
Furthermore, there should always remain enough of both cofibrations and fibrations, in
order to find functorial factorizations.
The small object argument is a generic tool that provides weak factorization systems,
given as input only a set of morphisms where the domain of each morphism is ’not too big’.
This requirement of objects being small enough is very important, since it is one of the only
problem that may occur. This machinery outputs two classes of maps I−cell and I−inj
such that any morphism of M can be factored by a map from I−cell followed by a map
from I−inj. The only missing property of (I−cell, I−inj) for being a weak factorization
system is that I−cell is not necessarily closed by retracts. Therefore, if we call I−cof the
closure by retracts of I−cell, the couple (I−cof, I−inj) is a weak factorization system, that
is, there is the functorial factorization required and every couple (i, p) ∈ (I−cof, I−inj)
satisfies the lifting property. This will be seen as one of the two factorization systems of a
model category.
However, this is only half of the model structure, since a model category is defined with
the two weak factorization systems (C , F ∩ W ) and (C ∩ W , F ). Since these two weak
factorization systems are certainly not independent, we could not just give as input two sets
of maps I and J and hope that the output gives a model structure on M. For example, a
relation between them may simply be the fact that every acyclic cofibration (i.e., an element
6
We will see that the realization functor preserves generating acyclic cofibrations and generating cofibrations, and therefore, preserves acyclic cofibrations and cofibrations.
49
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
of J −cof) is in particular a cofibration (i.e., an element of I−cof). The recognition theorem
will give a sufficient condition on the sets I and J in order to have an induced model
structure.
We will now give some more details and make precise this discussion. A complete
development can be found in Chapter 10 and 11 in [Hir03]. The functorial factorization of
a morphism X GGA Y will eventually be given by successively factoring it trough bigger
and bigger objects Zα for some (infinite) indexing α, until such an object Zβ is big enough
so that Zβ GGA Y has the right lifting property with respect to all desired maps. We need
first define such infinite compositions, and what is such a notion of ’size’ for objects in a
category.
Definition (λ-sequence, transfinite composition). Let λ be an ordinal, seen as a poset
category. A λ-sequence in a category M is a functor X : λ GGA M, i.e., a λ-diagram
X0 GGA X1 GGA · · · GGA Xβ GGA · · ·
∈M
∼
=
satisfying the property that the natural maps colimβ<γ Xβ GGA Xγ are isomorphisms for
all limit ordinals γ ≤ λ. The transfinite composition of a λ-sequence X is the morphism
X0 GGA colimβ<λ Xβ .
If D is a class of morphisms in M, a λ-sequence X in D is a λ-sequence X such that
every morphism Xβ GGA Xβ+1 lies in D.
∼
=
Intuitively, the condition colimβ<γ Xβ GGA Xγ for all limit ordinal γ is here to ensure
that X does not make gaps at these levels γ. Indeed, by their definition, limit ordinals
cannot be reached from below, and this condition is necessary in order to have some sort of
’continuity’ in a λ-sequence. These transfinite compositions allow the construction of very
big objects, by glueing objects together by means of pushouts. To control the size of the
objects that will arise in the functorial factorizations, we need to impose a condition on the
objects of (the domains of) I.
Definition (Regular cardinal). A cardinal λ is said to be a regular cardinal if for every set
S of cardinality less than λ and every collection of sets {Ss }s∈S such that each set Ss is of
cardinality less than λ, then the union ∪s∈S Ss is also of cardinality less than λ.
A regular cardinal can be seen as a limit ordinal that cannot be broken into a smaller
collection of smaller parts. We can now define what small objects are.
Definition ((κ-)small object (with respect to I)). Let I be a class of morphisms in Mor(M)
and κ be an ordinal. An object M ∈ M is said to be κ-small with respect to I if for every
regular cardinal λ ≥ κ and every λ-sequence
X0 GGA X1 GGA · · · GGA Xβ GGA · · ·
∈ M,
the induced function of hom-sets is a bijection
∼
=
colimβ<λ M(M, Xβ ) GGA M(M, colimβ<λ Xβ ).
50
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
An object M is said to be small with respect to I if there exists an ordinal κ such that M
is κ-small with respect to I. Moreover, the object M is said to be small if it small with
respect to the class of all morphisms Mor(M).
To best understand the definition, let’s consider a λ-sequence
X0 ,GGA X1 ,GGA · · · · · · ,GGA Xβ ,GGA · · · ,GGA colim Xβ
∈ M,
where all the maps are monomorphisms of M. This is in fact not really restrictive, it turns
out that it will often be the case that these maps are monomorphisms. In this case, the
induced function is injective
colim M(M, Xβ ) ,GGA M(M, colim Xβ ),
where the colimits are over β < λ. Indeed, observe that this function is just composition
with monomorphisms
X0
X1
···
···
Xβ
colim Xβ .
M
The bijectivity of this function is therefore equivalent to its surjectivity, which says that any
f
morphism M GGA colim Xβ admits a preimage, that is exactly to say that there is some
Xβ trough which f factors
X0
X1
···
···
Xβ
colim Xβ .
∃
f
M
This is essentially the meaning of the definition. In the category Set of sets, every set S is
card S-small. In the category sSet of simplicial sets, every simplicial set with a finite number
of non-degenerate simplices is ℵ0 -small with respect to all monomorphisms (cofibrations).
Similarly, in the category Top of topological spaces, any finite CW-complex is ℵ0 -small with
respect to the inclusions of CW-complexes.
Before stating the statement of the small object argument, we need to define the classes
of maps that will give the weak factorization systems.
Definition (I−inj, I−cof and I−cell). Let I be a set of morphisms in a cocomplete category M. Define three classes of morphisms by letting
• I−inj contains the morphisms that have the right lifting property with respect to all
morphisms in I;
51
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
• I−cof contains the morphisms that have the left lifting property with respect to all
morphisms in I−inj;
• I−cell contains all the morphisms that are obtained as transfinite compositions of
pushouts of coproducts of morphisms from I, that is, all the transfinite compositions
of the λ-sequences X in which each step Xβ GGA Xβ+1 is obtained as a pushout
a
`
Aβ,x
Xβ
fβ,x
a
p
Bβ,x
Xβ+1 ,
where each fβ,x ∈ I. A morphism A GGA B in I−cell is called a relative I-cell complex,
and an object X ∈ M is called an I-cell complex if the unique morphism ∅ GGA X is
a relative I-cell complex.
Definition. A set of morphisms I in M is said to permitting the small object argumentpermit the small object argument if the domains of each morphism in I are small with respect
to I.
Since a class of morphisms having the left (or right) lifting property with respect to another class is closed under transfinite compositions, I−cell ⊆ I−cof. The opposite inclusion
is not true, since I−cell is in general not closed under retracts, while I−cof is. In our case
of interest, when I permits the small object argument, this is the only obstruction and the
closure by retracts of I−cell gives exactly I−cof. The machinery that produces a weak
factorization is the following theorem.
Theorem 2.1.5 (The small object argument). Let M be a cocomplete category and I a
set of morphisms that permits the small object argument. Then there exists a functorial
factorization of every morphism of M by a morphism of I−cell followed by a morphism of
I−inj.
f
Sketch of the proof. Let X GGA Y be a morphism. The idea is to inductively (transfinitly)
factorizes it trough bigger and bigger objects, until the ’projection’ to Y admits the right
lifting property in all possible squares. More precisely, we will force this property to be true
by factorizing at each step ’trough all possible squares’. Graphically, the construction is the
following
52
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
X = X0
j1
f = p0
j2
X1
p1
···
X2
colim Xβ ,
p2
Y
p
where Xβ is a λ-sequence where each morphism is a pushout of coproducts of maps from I,
and the morphisms pβ are induced at each step by the universal property of the pushout.
pβ
Suppose the construction of Xβ GGA Y done, and let’s build Xβ+1 . Consider all possible
commutative squares
A
Xβ
pβ
i
B
Y,
where we require that i ∈ I. In order to have all the liftings required, we will formally
consider the pushout
a
`
dom i
Xβ
i
a
cod i
p
pβ
Xβ+1
pβ+1
Y
At first sight, we could proceed that way, and the algorithm may never stop, since at each
step there are new squares that we need to take care of. The trick comes from the fact
that all dom i that appear in these squares are all small. Therefore, each of them has an
associated ordinal κ such that for β ≥ κ, a morphism from dom i must factor trough Xβ .
Furthermore, by taking the supremum (union) of all these κ’s, we get an ordinal λ that has
the property that for any β ≥ λ, any map dom i GGA Xβ must factor trough Xλ , and so,
f
there are no new squares. All this is to say that if we consider the factorization of X GGA
53
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
at
j
p
X GGA colimβ<λ Xβ GGA Y,
the morphism j ∈ I−cell since it is a transfinite composition of pushouts of coproducts of
maps from I, and the morphism p grew enough to admit the right lifting property with
respect to all morphisms from I, as desired. For more details, see Proposition 10.5.16 in
[Hir03].
By closing the class I−cell under retracts, we get I−cof, which are exactly the maps
that have the left lifting property with respect to I−inj.
Corollary 2.1.6. Let M be a cocomplete category and I a set of morphisms that permits
the small object argument. Then the pair (I−cof, I−inj) is a weak factorization system on
M.
Proof. Since I−cof are exactly the morphisms that have the left lifting property with respect
to I−inj, and since I−cell ⊆ I−cof, the previous theorem gives the desired functorial
factorization.
This corollary gives a method of creating weak factorisation systems, and thus model
structures. A model category that can be obtain with two sets of maps I and J is called
cofibrantly generated.
Definition (Cofibrantly generated model category). A model category M with cofibrations
C , fibrations F and weak equivalences W is said to be cofibrantly generated if there exists
two sets of morphisms I and J , both permitting the small object argument, such that the
acyclic cofibrations are J −cof, the cofibrations are I−cof, the acyclic fibrations are I−inj
and the fibrations are J −inj. The set I is called the set of generating cofibrations and J is
called the set of generating acyclic cofibrations.
In a cofibrantly generated model category, the functorial factorization may not be the
one given by the small object argument, even this one is always available. Many usual model
categories are cofibrantly generated.
Example 2.2 (Cofibrantly generated model categories).
(1) The Quillen model structure on Top with weak homotopy equivalences and Serre fibrations is cofibrantly generated. The set I of generating cofibrations can be given
by the natural inclusions Sn ,GGA Dn+1 for n ∈ N, while the set of generating acyclic
cofibrations can be given be the inclusions Dn ,GGA Dn × I for all n ∈ N.
(2) The standard model structure on sSet with monomorphisms as cofibrations, Kan fibrations and weak equivalences the maps that are weak homotopy equivalences after
realization, is cofibrantly generated by letting the generating acyclic cofibrations being
Λk [n] ,GGA ∆[n] for 0 ≤ k ≤ n and the generating cofibrations ∂∆[n] ,GGA ∆[n] for
n ∈ N.
54
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
(3) The projective model structure on ChainR of chain complexes of R-modules, where
the cofibrations are the monomorphisms with projective cokernel, the fibrations are
the degree-wise epimorpisms and the weak equivalences are the quasi-isomorphisms, is
cofibrantly generated. This was proved by Hovey in Section 2.3 of [Hov99]. He denoted
Sn (R) to be the chain complex with R in degree n and 0 elsewhere, and Dn (R) to
be the chain complex with R in degree n and n − 1 with identity differential, and 0
elsewhere. He proved that the cofibrantly generated model structure defined by I to be
Sn ,GGA Dn+1 and J to be 0 GGA Dn for all integers n ∈ Z, is exactly the projective
model structure.
(4) In Section 2.1.3 of [Hov99], Hovey declares that possible exceptions are categories where
the weak equivalences are the strict homotopy equivalences. For example the category of chain complexes of abelian groups with chain homotopy equivalences as weak
equivalences, or the Hurewicz (or the Strom) model structure on Top with homotopy
equivalences as weak homotopy equivalences.
The dual notion of a fibrantly generated model category also makes sense, by letting
a model category being fibrantly generated if and only if its opposite model category is
cofibrantly generated. However, this notion is usually not relevant since the notion of
cosmall object is not very flexible. Indeed, even in the category Set of sets, the only cosmall
objects are the empty set and the singletons.
Cofibrantly generated model categories are useful for several reasons. First of all, it is
easier to endow a category with a model structure that is cofibrantly generated (Theorem
2.1.7, in particular because the small object argument gives the functorial factorization
(sufficient conditions are given in the next theorem). Moreover, carrying only part of the
data, by only keeping in mind the two sets I and J instead of C , F and W simplifies many
arguments. For example, it is now easier to verify when a functor F is a left Quillen functor
(Proposition 2.1.8), by only verifying it on the generating sets. With the same idea, it is
easier to transport a cofibrantly generated model structure to other categories. Important
examples that we will use throughout the project is the transport of a cofibrantly generated
model structure on M to categories of diagrams [C, M] (see Section3.1), categories that
GGA
have a reasonably good adjunction M DG
G ⊥ G N (Theorem 2.1.9) and last but not least,
localizations.
We start with the following theorem, sometimes called the recognition theorem, which
gives sufficient conditions such that two sets of maps I and J define a cofibrantly generated
model structure.
Theorem 2.1.7 (Recognition theorem, D.M.Kan). Let M be a complete and cocomplete
category with a class of maps W that is closed under retracts, and that have the 2-out-of-3
property. Let I and J be two sets of morphisms in Mor(M), that permit the small object
argument and satisfying
• J −cof ⊆ I−cof ∩ W ;
• I−inj ⊆ J −inj ∩ W ;
• either one of the two reverse inclusions.
55
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Then setting the cofibrations to be C = I−cof, the fibrations to be F = J −inj and the weak
equivalences to be W gives a cofibrantly generated model category with generating cofibrations
I and generating acyclic cofibrations J .
Proof. The proof is taken from Theorem 11.3.1 in [Hir03]. Observe that the conditions are
very intuitive. In words, the first one asks that an acyclic cofibration is both a cofibration
and a weak equivalence, and the second one is the similar statement for fibrations. The last
one asks for one of the reverse inclusion, even though both must hold.
The only axioms to be proved are the existence of the two functorial factorizations
and the lifting property of any acyclic cofibration with a fibration and of any cofibration
with an acyclic fibration. The functorial factorizations are both taken care by the small
object argument. For the lifting properties, suppose that we assume the hypothesis that
J −cof = I−cof ∩ W . Since (J −cof, J −inj) is a weak factorization system, the lifting
property of an acyclic cofibration and a fibration is verified. If we show the other inclusion
I−inj ⊇ J −inj ∩ W , then (I−cof, I−inj) is also a weak factorization system and we are
f
done. Let X GGA Y be a morphism in J −inj ∩ W . By the small object argument, we can
factorize it as
f
X
g
Y,
h
Z
where g ∈ I−cell ⊆ I−cof and h ∈ I−inj. By the 2-out-of-3 property of W , g ∈ W and
therefore by hypothesis g ∈ J −cof. But now, f is a retract of h and since I−inj is closed
by retracts, f ∈ I−inj, as desired. The proof assuming the other hypothesis is similar and
completely dual.
GGA
If F : M DG
G ⊥ G N : G is an adjunction between model categories, where M is cofibrantly
generated. The fact that F is a left Quillen functor (and thus that F a G is a Quillen
adjunction) can be checked on the sets of generating cofibrations I and the set of generating
acyclic cofibrations J of M.
GGA
Proposition 2.1.8. Let F : M DG
G ⊥ G N : G be a functor between a cofibrantly generated
model category M, with generating sets I and J , and any model category N . This adjunction is a Quillen adjunction if and only if F (f ) is a cofibration in N for all generating
cofibrations f ∈ I and F (f ) is an acyclic cofibration in N for all generating acyclic cofibrations f ∈ J .
Proof. This is taken from Lemma 2.1.20 in [Hov99]. The conditions are clearly necessary,
let’s show that they are also sufficient. It is easy to verify that F (I−cof) ⊆ F I − cof.
Furthermore, if we denote by CN the class of cofibrations of N , the hypothesis says that
F I ⊆ CN . It follows that F I − cof ⊆ CN − cof, but since CN − cof = CN , we have
F (I−cof) ⊆ F I − cof ⊆ CN − cof = CN ,
56
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
i.e., F sends cofibrations to cofibrations. Changing I by J gives that F sends acyclic
cofibrations to acyclic cofibrations, i.e., F is a left Quillen functor.
Another important result is that a cofibrantly generated model structure can be pushed
trough an adjunction under natural assumptions.
Theorem 2.1.9 (Kan). Let M be a cofibrantly generated model category with generating
cofibrations I and generating trivial cofibrations J . Let N be a complete and cocomplete
category and F : M N : G be an adjunction such that
(1) F I and F J permit the small object argument;
(2) G takes relative F J -cell complexes to weak equivalences, i.e., G(F J − cell) ⊆ WM .
Then N admits a cofibrantly generated model structure in which, F I is a set of generating
cofibrations, F J is a set of generating trivial cofibrations and the weak equivalences are the
maps that G takes to weak equivalences in M. Furthermore, the adjunction F a G is a
Quillen adjunction with respect to these model structures.
Proof. The proof is taken from Theorem 11.3.2 in [Hir03]. We will use the recognition
theorem and so the assumption that F I and F J permit the small object argument is
essential. Define the weak equivalences WN to be the morphisms that G takes to weak
equivalences in M. Since G preserves composition and retracts, the class WN is closed by
retracts and satisfies the 2-out-of-3 property.
By hypothesis F J −cell ⊆ WN and so by closing under retracts we get F J −cof ⊆ WN .
Moreover,
I−inj ⊆ J −inj
=⇒
F I−inj ⊆ F J −inj
=⇒
F J −cof ⊆ F I−cof,
and so the first hypothesis F J −cof ⊆ F I−cof ∩ WN of the recognition theorem is satisfied.
Since F I−inj ⊆ F J −inj by adjunction and using the lifting property we get that
G(F I−inj) ⊆ I−inj ⊆ WM . In particular this gives the second inclusion F I−inj ⊆
F J −inj ∩ WN .
f
For a reverse inclusion, let’s pick a morphism X GGA Y ∈ F J −inj ∩ WN . Again by
adjunction we get that U (f ) ∈ J −inj ∩ WM = I−inj, which gives the desired inclusion
F J −inj ∩ WN ⊆ F I−inj. By the recognition theorem 2.1.7, the sets F I and F J define a
cofibrantly generated model structure on N .
For the last point, observe that since F is a left adjoint, it preserves all colimits. In
particular F (I−cell) ⊆ F I−cell and F (J −cell) ⊆ F J −cell. Moreover, since any functor
preserves retracts, by closing under retracts these two inclusions, we get that F preserves
cofibrations as well as acyclic cofibrations, i.e., F is a left Quillen functor and so the adGGA
junction F : M DG
G ⊥ G N : G is a Quillen adjunction.
The strentgh of this theorem is that it suffices to prove that G sends the (regular) ’acyclic
cofibrations’ to weak equivalences, and the rest follows for free.
An example of an adjunction that allows such a lifting of a model structure is the
adjunction between a cofibrantly generated model category M and its pointed version.
57
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Given a category M with finite coproducts and with a terminal object ∗, its associated
pointed category, denoted by M∗ , is the category (∗ ↓ M) under the terminal object.
Moreover, there is an adjunction
GGA
(−)+ : M DG
G ⊥ G M∗ : U,
where U is the forgetful functor and (−)∗ adds a disjoint base point. More generally any
above or under category of M admits a natural model structure from M.
Theorem 2.1.10. Let M be a model category, and let M ∈ M be any object. Then the
categories (M ↓ M) and (M ↓ M ) admit a model structure in which a morphism is a
cofibration, a fibration or a weak equivalence if it is so in M.
Proof. Everything follows from the definitions.
In particular, there is a model structure on M∗ , where a map is a cofibration, a fibration
or a weak equivalence if it is so in M after applying the forgetful functor U . More precisely,
if M is cofibrantly generated, then so is M∗ .
Corollary 2.1.11. Let M be a cofibrantly generated model category with I as set of generating cofibrations and J as generating acyclic cofibrations. There is an induced cofibrantly
model structure on M∗ with generating cofibrations I+ and generating acyclic cofibrations
J+ , where a morphism f is a cofibration, fibration or weak equivalence if and only if U (f )
is respectively a cofibration, a fibration or a weak equivalence.
Proof. Consider the adjunction
GGA
(−)+ : M DG
G ⊥ G M∗ : U.
The model structure where the cofibrations, fibrations and weak equivalences of M∗ are
the ones that are so after applying the forget functor U turns M∗ into a model category.
The lifting property follows by the lifting property in M, which is lifted to M∗ trough the
adjunction, and the functorial factorization is similarly given by the one in M.
By adjointness, it follows that I+ −cof are the cofibrations, J+ −cof the acyclic cofibrations, J+ −inj are the fibrations, I+ −inj the acyclic fibrations. It remains to prove that I+
and J+ permit the small object argument in M∗ . Since the forget functor U commutes with
colimits of diagrams of the type
X0 GGA X1 GGA · · · GGA Xβ GGA · · · ,
the fact that I+ (or J+ ) permits the small object argument is the same as the fact that I
(or J ) permits the small object argument, which is true by assumption.
58
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
2.1.6
Cellular and combinatorial model categories
A cellular model category is essentially a cofibrantly model category, with two extra conditions, a stronger statement of smallness of objects and a condition on cofibrations. A
combinatorial model category requires an even stronger condition of smallness. These conditions are in particular here so that constructions, such as localization of model categories,
always exist. We start by defining the extra condition of smallness, which is a generalization
of a small object.
Definition (Compact object). Let I be a class of morphisms in a cocomplete category M
and κ be a cardinal. An object M ∈ M is said to be κ-compact with respect to I if for every
λ ≥ κ and any λ-sequence in I
X0 GGA X1 GGA · · · GGA Xβ GGA · · · GGA colim Xβ
∈ M,
the induced function of hom-sets is a bijection
∼
=
colimβ<λ M(M, Xβ ) GGA M(M, colimβ<λ Xβ )
The object M is said to be compact with respect to I if it is κ-compact with respect to I for
some κ, and it is said to be compact if it is compact with respect to all morphisms Mor(M).
By considering the covariant hom-functor M(M, −) : M GGA Set, the fact that M is
κ-compact is to say that it preserves all the colimits colimβ<λ Xβ for all λ ≥ κ. Recall that
an object M was defined to be κ-small if it preserves all these colimits for only the regular
cardinals λ ≥ κ.
Example 2.3 (Compact objects in sSet). This is our foundational example, since our main
categories will be diagrams in sSet. Any finite simplicial set K• ∈ sSet, i.e., that only has
a finite number of non-degenerate simplices, is ω-compact, where ω is the countable infinite.
f
In fact consider an ω-sequence X and a map K• GGA colim X ∈ sSet
X0
X1
Xn
···
f
···
colimn∈N X n .
∃
K•
Since there are only a finite number of non-degenerate simplices and the colimit is X n / ∼,
there is a sufficiently large natural number n such that X n contains the image of all the
non-degenerate simplices of K• . That is to say f (K• ) ⊆ X n and thus f factorizes through
X n.
`
We will now define the condition imposed on cofibrations, that our favourite model
category sSet satisfies on the nose.
59
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
i
Definition (Effective monomorphism). In a category M, a morphism K GGA K is said to
be an effective monomorphism if the pushout L
`
GGA L K L.
of L GGA
i
`
K
L exists, and if K GGA is the equalizer
If we construct the pushout square
i
K
L
i
pL
L
`
K
L,
i
saying that K GGA L is the desired equalizer, is the same as requiring a unique filler in
A
j
∃!
j
i
K
L
i
pL
L
j
for any map A GGA K that gets equalized in L
`
K
`
K
L,
L.
Example 2.4 (Effective monomorphisms in Set and sSet). Let’s first show that effective
i
monomorphisms are exactly the injections in the category Set of sets. If K GGA L is
j
`
GGA L K L, then
an injection, and if A GGA L is equalized by the induced pair L GGA
j(A) ⊆ i(K) ⊆ L. Therefore such a filler A GGA K exists and is forced to send an element
a G
[ GA i−1 ◦ j(a).
More generally, an effective monomorphisms is necessary a monomorphism. Indeed, pick
i
GGA K that are the same after postcomposition with K GGA L,
two morphisms f, g : A GGA
`
GGA L K L, and
i.e., i ◦ f = i ◦ g. They are therefore the same after composition with L GGA
i
by the property of K GGA K being an effective monomorphism, there is only one such
i
A GGA K, i.e., f = g. This shows that the morphism K GGA L is a monomorphism.
i
In the category sSet of simplicial sets, let K• ,GGA L• be a monomorphism. Since
colimits are computed degree-wise, we can restrict our attention to diagrams
60
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
An
jn
∃?
in
Kn
jn
Ln
in
p
Ln
Ln
`
Kn
Ln ,
of sets. Since monomorphisms of simplicial sets are degree-wise injections (of sets), the
argument shows that there is a filler, and it is unique. Doing the argument in each degree
i
show that K• ,GGA L• is an effective monomorphism.
Definition (Cellular model category). A cofibrantly generated model category M is said
to be cellular if there is a set I of generating cofibrations and a set J of generating acyclic
cofibrations such that
• the domains and codomains of morphisms in I are compact objects;
• the domains of morphisms in J are small with respect to I;
• the cofibrations (given by I−cof) are effective monomorphisms.
Suppose given a model category M that is cofibrantly generated with set I of generating
cofibrations and set J if generating acyclic cofibrations. If the model category M is cellular,
the generating sets Ie and Je of the cellular structure need not be directly related to I and
J (as sets). Of course, since the cellular structure has the same underlying model structure
as the given structure on M, relations such as I−cof = Ie − cof must hold.
Example 2.5 (Cellular model category).
(1) The usual structure on sSet, where the set I of generating cofibrations is given by
I = {∂∆[n] ,GGA ∆[n]}n∈N ,
the set J of generating acyclic cofibrations is given by
n
o
J = Λk [n] ,GGA ∆[n]
k≤n ,
n>0
is a cellular model structure. The cofibrations are (exactly all the) effective monomorphisms since they are exactly the monomorphisms. Moreover, all the domains and
codomains appearing both in I and J are compact (with respect to all morphisms)
since they are finite. In particular, this model structure is cellular.
(2) Similarly, the category sSet∗ of pointed simplicial sets is also a cellular model category.
61
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
The second type of model category is a combinatorial model category. As its name
indicates, it is combinatorial in the sense that all objects are built up (as colimits) from
smaller objects, as is for example the case in sSet.
Definition (Locally presentable category). A cocomplete category M is called a locally
presentable category if
• all objects are small;
• there exists a set S of objects of M such that each object of M can be obtained as a
colimit of a diagram with only objects of S.
To emphasize the size of the objects of S, a locally presentable category M where the
set S may be chosen among κ-small objects (for a regular cardinal κ) is said to be κ-locally
presentable. If we choose κ = ω, the countable infinite, an ω-locally presentable category is
called locally finitely presentable.
Example 2.6 (Locally presentable categories).
(1) The category Set of sets is locally finitely presentable, since any set X ∈ Set is the
(directed) colimit of the poset (under inclusion ⊆) of its finite subsets. Therefore, the
set of its generators contains one set of n elements for any natural number n ∈ N.
(2) The category sSet of simplicial sets is also locally finitely presentable since a simplicial
set K• is a colimit over its category of simplices. More precisely, construct the category
of simplices of K• , denoted by ∆K•
• Objects : morphisms ∆[n] GGA K• ∈ sSet, for any n ∈ N;
f
• Morphisms : morphisms between the domains ∆[n] GGA ∆[m] over K• , i.e., such
that the diagram commutes
f
∆[n]
∆[m]
K• .
The composite of the projection functor
π : ∆K• GGA ∆ : (∆[n] → K• ) G
[ GA [n],
with the Yoneda embedding
∆ ,GGA [∆op , Set] = sSet : [n] G
[ GA ∆[n],
gives the chain of functors
∆K• GGA ∆ ,GGA sSet : (∆[n] → K• G
[ GA ∆[n].
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CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
The simplicial set K• is canonically isomorphic to the colimit of this diagram in sSet
colim∆K• ∆[n] ∼
= K•
naturally in the category sSet.
See Proposition 3.2.1 for a proof of a generalization of this result. Therefore, the category
sSet of simplicial sets is locally finitely presentable, generated by the representables
S = {∆[n]}n∈N .
(3) If C is a small category, the category of presheaves [C op , Set] is also finitely locally
presentable. This is a generalization of the above result, proved in Proposition 3.2.1.
Definition (Combinatorial model category). A model category M is called combinatorial
if it is locally presentable and cofibrantly generated.
Example 2.7 (Combinatorial model categories). The model structure on sSet is combinatorial, since it is cofibrantly generated and the underlying category is locally finitely presentable. We will see that the standard models on the categories of simplicial preseaves
[C op , sSet], the important categories that we will study, are also combinatorial.
Locally presentable categories are very useful in homotopy theory because, since all
objects are small, we can freely apply the small object argument which helps creating model
structures.
2.1.7
Proper model categories
Properness is another useful property that model categories may enjoy. It follows from their
lifting properties that cofibrations are preserved by pushouts and fibrations are preserved
by pullbacks, as in
A
X
A
∴
∴
pY
B
X
y
B
Y.
Definition (Left/right proper model category). A model category M is said to be
• left proper if the pushout of a weak equivalence along a cofibration is again a weak
equivalence (the diagram on the left);
• right proper if the pullback of a weak equivalence along a fibration is again a weak
equivalence (the diagram on the right);
• proper if it is both left and right proper.
A
B
∼
∴∼
X
A
pY
B
63
∴∼
y
∼
X
Y.
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
The axioms of a model category imply that the pushout along a cofibration of a weak
equivalence between cofibrant objects, is necessary a weak equivalence, and dually for pullbacks along fibrations.
Proposition 2.1.12 (Reedy). Let M be a model category.
• Every pushout of a weak equivalence between cofibrant objects, along a cofibration, is
again a weak equivalence.
• Every pullback of a weak equivalence between fibrant objects, along a fibration, is again
a weak equivalence.
Proof. The orignial proof is due to Reedy in the article [Ree74]. The proof may also be
found as Proposition 13.1.2 in [Hir03].
Corollary 2.1.13. Let M be a model category. If every object is
• cofibrant, then M is left proper.
• fibrant, then M is right proper.
• bifibrant, then M is proper.
Proof. This follows immediately from the last proposition.
Example 2.8 (Left/right proper model categories).
(1)
(2)
(3)
(4)
The model structure of Quillen on Top is right proper since all objects are fibrant.
The model structure of Hurewicz (Strom) on Top is proper since all objects are bifibrant.
The usual model structure on sSet is left proper since all objects are cofibrant.
In the case of the category of chain complexes, for similar reasons, the injective model
structure is left proper while the projective model structure is right proper.
(5) The pointed versions of Top and sSet have the same type of properness as their unpointed versions.
In fact, the model structures on Top and on sSet are both proper. We will need in
particular this result for the model category sSet of simplicial sets.
Proposition 2.1.14. The usual model structure on sSet and on its pointed version sSet∗
are both proper.
Proof. The proof is taken from Theorem 13.1.13 in [Hir03]. Their left properness follows
from the fact that every object is cofibrant. Consider now the realization-singular adjunction
GGA
Re : sSet DG
G ⊥ G Top : Sing.
The right properness will follow from the right properness of Top and Top∗ and from the
two facts that
• the realization functor commutes with finite limits;
64
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
• the geometric realization of a Kan fibration is a Serre fibration, i.e., a fibration in the
Quillen model on Top.
The second fact was originally proved by Quillen in [Qui68]. Using the first fact, the realization of a pullback square in sSet is a pullback square in Top. The result follows by using
the right properness property in Top since a morphism X GGA Y is a weak equivalence in
sSet if its realization is a weak equivalence in Top.
The proof that Top is proper can be found as Theorem 13.1.11 in [Hir03], where all the
Chapter 13 is devoted to the study of proper model categories.
2.2
Simplicial Model Categories
We will now consider another very useful type of model categories : simplicial model categories. Simplicial categories (not necessarily a model category) admit a natural mapping
space Map(A, B) ∈ sSet between any two objects A, B ∈ M. This is very useful in a simplicial model category, because the compatibility axiom that they are required to satisfy gives
a homotopy theoretic meaning to such a mapping space. This section is highly inspired by
Chapter 2 in [GJ09].
2.2.1
Simplicial categories
Definition (Simplicial category). A simplicial category M is a category enriched over the
standard symmetric closed monoidal structure of sSet.
The monoidal structure on sSet is tensored with the cartesian product × and cotensored
with the internal-hom
sSet(K• , L• ) : n G
[ GA sSet(K• , L• )n := sSet(K• × ∆[n], L• ).
An enriched structure on sSet is given by
• a class of objects Ob(M), also denoted by M;
• for every pair of objects A, B ∈ M a simplicial set that we will denote by Map(A, B) ∈
sSet and call the (simplicial) mapping space between A and B;
• for every triple of objects A, B, C ∈ M, a composition morphism
Map(A, B) × Map(B, C) GGA Map(A, C)
∈ sSet
that is associative;
• for every object A ∈ M, a unit morphism
∗ GGA Map(A, A)
∈ sSet,
that satisfies the natural unit coherence with respect to composition.
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CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
All the enrichments that we will consider will be tensored and cotensored, that is, for
every object A ∈ M and every simplicial set K• ∈ sSet there are tensors and cotensors
K• ⊗ A ∈ M
and
AK• ∈ M,
with natural isomorphisms of simplicial sets
Map(K• ⊗ A, B) ∼
= sSet(K• , Map(A, B)) ∼
= Map(A, B K• )
∈ sSet.
To avoid undefined notions and incoherences, since all the structures we will deal with are
symmetric, let’s also define A ⊗ K• := K• ⊗ A. If M has an underlying category, which will
always be the case in all our considerations, we recover the sets of morphisms of M by
?
M(A, B) ∼
= sSet(∗, Map(A, B)) ∼
= Map(A, B)0
∈ sSet
where (?) is given by the discussion after Proposition 1.2.1. From these two chains of three
isomorphisms, all the possible coherent adjunctions with tensors and cotensors between M
and sSet hold. For example, given an object A ∈ M, the adjunction
GGA
− ⊗ A : sSet DG
G ⊥ G M : Map(A, −)
is just the 0th level of the tensoring isomorphisms Map(K• ⊗ A, B) ∼
= sSet(K• , Map(A, B)).
The adjunction
op
GGA
A(−) : sSet DG
G ⊥ G M : Map(−, A)
is just the 0th level of the (contravariant version) of the cotensoring isomorphism
sSet(K• , Map(A, B)) ∼
= Map(A, B K• )
⇐⇒
Mapop (B K• , A) ∼
= sSet(K• , Map(A, B)).
If we now fix a simplicial set K• we also have the adjunction
K•
GGA
− ⊗ K• : M DG
G ⊥ G M : (−) .
Another really important remark is the interpretation of the n-simplices of the mapping
space Map(A, B). By first using the Yoneda lemma and then one of the above adjunctions,
we get
Map(A, B)n ∼
= sSet(∆[n], Map(A, B)) ∼
= M(A ⊗ ∆[n], B).
This is an important fact because we can now understand the mapping space only by
looking at the underlying category M. In an ideal world, every model category should
have a notion of simplicial mapping space, which moreover is required to have a homotopy
theoretic content, i.e., it is a simplicial model category. Hopefully, all our categories of
interest will be of this kind, inheriting the properties of sSet.
Example 2.9 (Tensored and cotensored sSet-enriched categories).
(1) Simplicial sets sSet is a tensored and cotensored sSet-category, with simplicial mapping
space given by
Map(K• , L• ) := sSet(K• , L• ) = sSet(K• × ∆[−], L• ).
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CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
(2) Pointed simplicial sets sSet∗ is also a tensored and cotensored sSet-category. We will
use the standard adjunction
GGA
(−)+ : sSet DG
G ⊥ G sSet∗ : U,
where K+ := ∗ K and U forgets about the base point, and the smash product of
pointed simplicial sets is
`
− ∧ − : sSet∗ × sSet∗ GGA sSet∗ : (X, Y ) G
[ GA X ∧ Y := X × Y /X ∨ Y,
where × is the product in sSet∗ and ∨ is the coproduct (wedge product). The tensor
and cotensor of K ∈ sSet and a pointed X∗ ∈ sSet is
K ⊗ X∗ := K+ ∧ X∗
∈ sSet∗
and
X∗K := sSet(K, U (X∗ ))
∈ sSet∗
that is pointed by the constant map K GGA ∗U (X∗ ) sending everything to the base
point. The simplicial (unpointed !) mapping space between two pointed simplicial sets
X∗ , Y∗ ∈ sSet∗ is
Map(X∗ , Y∗ ) = sSet(U (X∗ ), U (Y∗ )) ∼
= sSet(U (X∗ ∧ ∆[−]+ ), U (Y∗ )),
which is the same as the unpoited version, just using − ∧ ∆[n]+ instead of − × ∆[n].
(3) Let C be a small category. The category of simplicial presheaves on C, which is the
category of functors M := [C op , sSet] is a tensored and cotensored simplicial category.
There is an embedding of sSet into M, by sending a simplicial set K• to the constant
simplicial presheaf
f• : C op GGA sSet : C G
K
[ GA K• .
The tensor of K• ∈ sSet with a simplicial presheaf F ∈ M is given by the pointwise
product in M after embedding K• in M
f• × F : C G
K• ⊗ F := K
[ GA K• × F (C),
and similarly the cotensor is given by
e
F K• := F K• : C G
[ GA F (C)K• = sSet(K• , F (C)).
Therefore, there is a notion of simplicial mapping spaces in M and, for two simplicial
presheaves F, G ∈ M it is given by the simplicial set
e
Map(F, G) : n G
[ GA sSet(F × ∆[n],
G).
2.2.2
Simplicial model categories
If M is a cotensored and tensored sSet-enriched category that admits a model structure
(C , F , W ), we would like to add a homotopic meaning to the simplicial mapping spaces
Map(A, B) ∈ sSet.
67
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Definition (Simplicial model category). A simplicial model category is a model category
M that also is a sSet-enriched category, satisfying the axioms that
(SM6) it is a tensored and cotensored as a sSet-category;
p
i
(SM7) for every cofibration A GA B ∈ C and any fibration X GGA
A Y ∈ F , the induced
morphism of mapping spaces
(i∗ ,p∗ )
Map(B, X) GGGGA Map(A, X) ×Map(A,Y ) Map(B, Y ) ∈ sSet,
is a fibration of simplicial sets, which is a weak equivalence if either i or p is.
The tensored and cotensored conditions are here to ensure that there is a notion of simplicial
mapping space Map(X, Y ) ∈ sSet between any two objects in M. Within the framework of
this project, the existence of such a mapping space is our main interest for simplicial model
categories. The morphism (i∗ , p∗ ) that is precomposition with i and postcomposition with
f
p, i.e., can be illustrated as sending morphism B GGA X to a commutative square in which
it is a filler
A
i
B
X
f
p
A
=⇒
i
Y
B
f ◦i
X
f
p
Y.
p◦f
A first homotopic consequence of this axiom are the following observations on the mapping spaces Map(−, −).
p
Lemma 2.2.1. If A is cofibrant and X GGA
A Y is a fibration, then the induced map
p∗
Map(A, X) GGA
A Map(A, Y )
i
is also a fibration, which is a weak equivalence if p is. Dually if A GA B is a cofibration,
then
i∗
Map(B, Y ) GA Map(A, Y )
is also a cofibration, which is a weak equivalence if i is.
Proof. This is the definition applied to the cofibration ∅ GA A and the (acyclic) fibration
p
i
X GGA
A Y for the first, and to the cofibration A GA B and the (acyclic) fibration Y GGA
A∗
for the second.
Corollary 2.2.2. If B is cofibrant and X is fibrant, then Map(B, X) is a fibrant simplicial
set, i.e., a Kan complex.
68
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
∼
Corollary 2.2.3. If B is cofibrant and X GGA Y is a weak equivalence of fibrant objects,
then so is
∼
Map(B, X) GGA Map(B, Y ).
∼
Dually, if X is cofibrant and A GGA B is a weak equivalence of cofibrant objects, then so is
∼
Map(B, X) GGA Map(A, X).
Another interpretation of this axiom is that it strengthens axiom (CM4) about the lifting
properties of the pairs (C , F ∩ W ) and (C ∩ W , F ).
Proposition 2.2.4. Let M be a simplicial category with three classes C , F and W (but
not necessarily a model category) satisfying axioms (SM6) and (SM7), and consider a commutative square
A
X
p
i
B
Y,
p
i
in which A GGA B is a cofibration and X GGA Y is a fibration. If either i or p is a weak
f
equivalence, then there exists a diagonal filler B GGA X.
Proof. Given the morphisms i and p, such a commutative square is a 0-simplex in Map(A, X)×Map(A,Y )
Map(B, Y ). Axiom (SM7) implies that
(i∗ ,p∗ )
Map(B, X) GGGGA Map(A, X) ×Map(A,Y ) Map(B, Y ) ∈ sSet
is a trivial fibration, i.e., that it is surjective. Therefore, any such square admits a
preimage in Map(B, X), i.e., a diagonal filler as desired.
In fact, even more is true in a simplicial model category. Such a lifting
A
i
X
f
p
B
Y
is unique up to homotopy under A and above Y . See Proposition 3.8 in [GJ09] for more
explanations and a proof of this statement.
It is often hard to come up with a direct proof that a model category that is also tensored
and cotensored as a sSet-category is simplicial. Even showing that the category sSet itself
69
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
is a simplicial model category requires some technical work, see for example Section 3.3 in
[Hov99]. We will rely on this fact to prove that our categories of interest are simplicial, and
that most of the constructions on them preserve this property (taking diagrams on them or
localizing, for example). By using all the adjunctions developed in the simplicial categories,
there are two useful characterizations of axiom (SM7).
Proposition 2.2.5. In a tensored and cotensored simplicial category M, for any morphisms
i
p
j
A GGA B, X GGA Y ∈ Mor(M) and any map K GGA L ∈ sSet, the three following
statements are equivalent
(1) For any commutative square in the category sSet there is a diagonal filler
K
Map(B, X)
∃f1
j
(i∗ , p∗ )
Map(A, X) ×Map(A,Y ) Map(B, Y ).
L
(2) For any commutative square in the category M there is a diagonal filler
a
K ⊗B
L⊗A
K⊗A
i
X
∃f2
p
j
L⊗B
Y,
where i j is the natural morphism, that only depends on i and j.
(3) For any commutative square in the category M there is a diagonal filler
XL
A
i
B
where p
∃f3
p
i
Y L ×Y K X K .
i is the natural morphism, that only depends on p and i.
Proof. Observe that there is a special diagram for each one of the three maps i,j and p is
represented, and where the other vertical map only depends on the two other maps. The
key adjunctions are the three possible natural isomorphisms of sets
sSet(L, Map(B, X)) ∼
= M(L ⊗ B, X) ∼
= M(B, X L ).
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CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
These isomorphisms are also the ones that gives the relations between the three fillers f1 ,
f2 and f3 . The first diagram may for example be refined by developing the pullback and
the three important maps in this diagram are encoded in the two horizontal maps, which
give induced maps
k1
l1
K GGA Map(B, X),
L GGA Map(A, X)
m1
L GGA Map(B, Y ),
and
and they must satisfy relations, being equal by all the possible ways to arrive at Map(A, Y ).
Trough the adjunctions, they give raise to the three maps for (2)
k2
K ⊗ B GGA X,
l2
L ⊗ A GGA X
m2
L ⊗ B GGA Y,
and
or to the three maps for (3)
k3
B GGA X K ,
l3
A GGA X L
and
m3
B GGA Y L .
These adjuncions give the diagrams (2) and (3), and the fillers f1 , f2 and f3 are related by
the natural isomorphisms
f1 ∈
z
}|
f3 ∈
f2 ∈
{
z
}|
{
z
}|
{
sSet(L, Map(B, X)) ∼
= M(L ⊗ B, X) ∼
= M(B, X L ) .
This gives alternative possibilities for checking the axiom (SM7), that we will use to prove
that the category of simplicial presheaves is a simplicial model category.
2.3
Localization of Model Categories
This is a very vast subject that is treated in great details in the book [Hir03] of Hirschhorn,
and we will only give a very brief overview. Throughout this section, we will assume that
all the model categories have a notion of simplicial mapping space Map(−, −). This is not
really a restriction for two reasons
(1) all our categories will be simplicial, and so Map(A, B) := M(A ⊗ ∆[−], B) plays this
role;
(2) all model categories, even the ones that are not enriched over sSet, admit a notion of
simplicial mapping space, usually called a homotopy function complex, this is treated in
Chapter 17 of [Hir03] for example.
Moreover, we will assume that Map(−, −) is a bifuncto Mop × M GGA sSet, which it is
both with our definition Map(A, B) := M(A ⊗ ∆[−], B) in the case of a simplicial model
category, and also more generally as a homotopy function complex in any model category.
An important role of a model category is to serve as a presentation of its homotopy
category, which is roughly obtained by quotienting the hom-sets after a cofibrant and fibrant
replacement. Conceptually, all the general definitions, results, constructions that are ’up to
isomorphism’ in the homotopy category Ho(M), happens ’up to homotopy’ in the model
71
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
category M. For example, a morphism f ∈ Mor(M) is a weak equivalence if and only if its
image [f ] ∈ Mor(Ho(M)) is an isomorphism.
Following this idea, a localization of a model category M with respect to a class of maps
S ⊆ Mor(M) should therefore be another (universal) model category MS , related to the
l
orignial category by some localization morphism M GGA MS , such that the image of all
morphisms of l(S) are isomorphisms in the homotopy category Ho(MS ). Roughly speaking,
all the morphisms in l(S) will be weak equivalences in MS , as well as all the morphisms that
are forced to be weak equivalences if all the morphisms of l(S) are, for example composites
of morphisms from l(S). Moreover, since there are two notions of morphisms between model
categories, the left and right Quillen functors, there will be two notions of localization of
model categories.
Definition (Left/right localization of a model category). Let M be a model category and
S ⊆ Mor(M) any class of morphisms. A left localization of M with respect to S is a model
j
category LS M with a left Quillen functor M GGA LS M such that
• the total left derived functor Lj : Ho(M) GGA Ho(LS M) sends the image in the homotopy category of all morphisms of S, to isomorphisms in Ho(LS M);
• it is universal for this property, i.e., for any other left Quillen functor j 0 : M GGA N
that satisfies the property, there is a unique left Quillen functor σ : LS M GGA N such
that the diagram commutes
M
j0
j
LS M
N.
∃! σ
Dually, a right localization of M with respect to S is a model category RS M with a right
j
Quillen functor M GGA RS M such that
• the total right derived functor Rj : Ho(M) GGA Ho(RS M) sends the image in the
homotopy category of all morphisms of S, to isomorphisms in Ho(RS M);
• it is universal for this property, i.e., for any other right Quillen functor j 0 : M GGA N
that satisfies the property, there is a unique right Quillen functor σ : RS M GGA N such
that the diagram commutes
M
j0
j
RS M
∃! σ
N.
Since they are defined with a universal property, if a left or a right localization exist,
then it is unique up to unique isomorphism. There are two large classes of model categories
72
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
for which it is known how to construct a localization with respect to a set of maps (it is
important that it is not a proper class), they are
• the left proper cellular model categories (treated in [Hir03]);
• the left proper combinatorial model categories (this is due to J. Smith).
At least in the case of left proper cellular model categories, there is a very explicit construction of such a localization. More precisely, a localization of such a category M with
respect to a set of maps S, will be a different model structure LS M or RS M on the same
underlying category M, with in addition Quillen adjunctiown that are given by the identity
functors
GGA
GGA
id : M DG
or
id RS M DG
G ⊥ G LS M : id
G ⊥ G M : id .
This particular localization is called a left/right Bousfield localization. We will present
the version of the left Bousfield localization and refer to Chapter 5 in [Hir03] for the right
Bousfield localization, that we won’t use. Defining, the new model structure LS M on the
underlying category of M is done by first seeing which objects will be forced to be fibrant
in the new structure, and also which maps will be forced into weak equivalences.
Definition (S-local objects, S-local equivalences). Let M be a model category and S ⊆
Mor(M) a class of morphisms. An object M ∈ M is said to be S-local if it is fibrant in M
f
and if for any morphism A GGA B ∈ S, the induced map of mapping spaces
∼
f ∗ : Map(B, M ) GGA Map(A, M )
∈ sSet
is a weak equivalence of simplcial sets.
f
A morphism X GGA Y ∈ Mor(M) is said to be an S-local equivalence if for every S-local
object M , the induced map of mapping spaces
∼
f ∗ : Map(Y, M ) GGA Map(X, M )
∈ sSet
is a weak equivalence of simplcial sets.
This definition says that the S-local objects are the fibrant objects that see every morphism in S as a weak equivalence, and the S-local equivalences are the morphisms that are
seen as weak equivalences by every S-local object. In fact, the S-local objects will exactly
be the fibrant objects (that is contained in the fibrant objects of M) in the new model
structure, and the S-local equivalences will exactly be the weak equivalences (that contains
the weak equivalences of M).
Definition (Left Bousfield localization). Let M be a model category S ⊆ Mor(M) a class
of morphisms. A left Bousfield localization of M with respect to S is another model structure
on M, denoted by LS M, where
• the cofibrations are the same as in M;
• the weak equivalences are the S-local equivalences;
73
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
• the fibrations are the morphisms with the right lifting property with respect to all
cofibrations that are also S-local equivalences.
This definition does not say that for M a model category and S ⊆ Mor(M) a class of
morphisms, a left Bousfield localization exists. Even though there is no existential problems
for three classes of morphisms in M, they may very well not form a model structure on M,
i.e., they may not satisfy the 5 axioms of a model category. Moreover, this definition does
not even assert that a left Bousfield localization is either a left or a right localization, which
we will have to prove.
For convenience, let’s call a hypothetical fibration in LS M, i.e., a morphism that has
the right lifting property with respect to all cofibrations that also are S-local equivalences,
an S-fibration. Similarly, let’s call a cofibration that also is an S-local equivalence, an Sacyclic cofibration. Observe that since the cofibrations are not changed, the hypothetical
class of acyclic fibrations in LS M is exactly the class of acyclic fibrations in M. Using
cofibrant approximations and simplicial resolutions, it is always true that the class of Slocal equivalences satisfies the 2-out-of-3 property and is closed by retracts. Moreover, the
cofibrations are already closed by retracts, as well as the S-fibrations, since they are exactly
the maps that have the right lifting property with respect to some class of maps. Moreover,
since the classes of cofibrations and acyclic fibrations are the same, half of (MC4) and half of
(MC5) are automatically verified. Therefore, to prove that such a left Bousfield localization
exists, it only remains half of axiom (MC4) and half of (MC5) to be proved. The task
however can not be done directly, and the proof relies heavily on the fact that the initial
model category M is cofibrantly generated.
Theorem 2.3.1 (Existence of left Bousfield localization). Let M be a left proper cellular
model category, and let S be a set (not a a proper class) of morphisms in Mor(M). The left
Bousfield localization LS M exists, i.e., there is a model structure on the underlying category
M where
• the cofibration are the same as in M;
• the weak equivalences are the S-local equivalences;
• the fibrations are what they are forced to be, i.e., the morphisms with the right lifting
property with respect to all cofibrations that are S-local equivalences.
Moreover, LS M is again a left proper cellular model category, and the fibrant objects are
exactly the S-local objects.
In addition, if M is also a simplicial model category, then this induces a structure of a
simplicial model category on the left Bousfield localization LS M.
General idea of the proof, taken from Section 4 in [Hir03]. Let M be a left proper cellular
model category, with generating sets (for the cofibrantly generated structure) I for the
cofibrations and J for the acyclic cofibrations. The set I generates the acyclic fibrations
and the cofibrations of both M and LS M, by respectively I−inj and I−cof. The hard part
is to find a set JS that generates fibrations and acyclic cofibrations.
74
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Fact (Section 4.5 in [Hir03]). There is a set JS of inclusions of cell complexes such that the
class JS −cof are exactly the cofibrations that are also S-local equivalences
The proof uses a size argument, which is called the Bousfield-Smith cardinality argument.
The control that we have on the domains of this set JS is knowing that the domains are
small, but we can’t prove yet that it permits the small object argument. The hypothesis of
a cellular model category is crucially used at this point.
Fact (Theorem 12.4.3 in [Hir03]). In a cellular model category, every cofibrant object is
small relative to the subcategory of cofibrations.
Therefore, the domains of the set JS are small relative to I, and therefore relative to
JS . We can now hope to apply the recognition theorem 2.1.7. The first inclusion JS − cof ⊆
I−cof ∩ WS is given by the following lemma.
Fact (Lemma 4.5.2 in [Hir03]). Let M be a left proper cellular model category and S ⊆
Mor(M) a set of morphisms. If p : E GGA B is a fibration with the right lifting property
with respect to all inclusions of cell complexes that are S-local equivalences, then it has the
right lifting property with respect to all cofibrations that are S-local equivalences.
For the other inclusion, since JS − cof ⊆ I−cof, then I−inj ⊆ JS − inj, and since
I−inj ⊆ W ⊆ WS , the second inclusion I−inj ⊆ JS − inj ∩ WS holds.
Finally, the reverse inclusion JS − cof ⊇ I−cof ∩ WS holds from the first fact, and so
Theorem 2.1.7 shows that LS M is indeed a cofibrantly generated model category.
Proving that the fibrant objects in LS M are exactly the S-local objects does not require
the cellular hypothesis, and could have been done before, for example Proposition 3.4.1 in
[Hir03].
Since the cofibrations of M are the same as the cofibrations of LS M, then the same set
of generating cofibrations for the cellular model on M, which is a set whose domains and
codomains are compact objects with respect to I, is also a candidate for a generating set
of cofibrations for a cellular structure on LS M. Moreover, the fact that the domains of the
object of J are small with respect to I is satisfied by the above Theorem 12.4.3. Finally,
since the underlying categories of M and LS M are the same, and since they have the same
cofibrations, it follows that every cofibration in LS M is an effective monomorphism, and
so LS M is cellular. Using general properties of homotopy function complexes, Proposition
3.4.4 in [Hir03] shows that if M is left proper (don’t need the cellular condition), then any
left Bousfield localization LS M is also left proper.
For the last claim, let’s assume that M is indeed a simplicial model category. In particular, the underlying category of M, which also is the underlying category of LS M, is
tensored and cotensored as an sSet-enriched category. Therefore, they share a same simplii
cial mapping space Map(−, −). For the axiom (SM7) in LS M, let A GA B be a cofibration
p
and let X GGA
A Y be a fibration in LS M. In particular, p is also a fibration in M and so
the axiom (SM7) of M implies that the map
Map(i, p) : Map(B, X) GGGA Map(A, X) ×Map(A,Y ) Map(B, Y )
∈ sSet
is a Kan fibration. If p is also an S-local equivalence, the it is an acyclic fibration in LS M,
i.e., also in M, and so the map Map(i, p) is an acyclic fibration. If i is also an S-local
75
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
equivalence, the proof is much longer, involving cosimplicial resolution and general notions
of a homotopy function complex, see Section 4.6 in [Hir03].
Examples of left proper cellular model categories, that admit left Bousfield localizations
for any set of maps S, are mainly sSet, Top, sSet∗ , Top∗ , as well as categories [C, M]
of C-diagrams in a left proper cellular model category M (the model structure is given in
Section 3.1).
To see that a left Bousfield localization, when it exists, is a left localization, we first need
a left Quillen functor j : M GGA LS M whose total left derived functor Lj : Ho(M) GGA
Ho(LS M) sends the image of S to isomorphisms. The identity functor
id : M GGGA LS M
is a left Quillen functor since it trivially preserves cofibrations, and also acyclic cofibrations
since weak equivalences are S-local equivalences. Since the right adjoint is again the identity
functor, this extends into an adjoint pair of identity functors
GGA
id : M DG
G ⊥ G LS M : id,
which is a Quillen adjunction. Moreover, since for any S-local object M ∈ M, any morphism
∼
A GGA B ∈ S induces a weak equivalence of simplicial sets Map(B, M ) GGA Map(A, M ),
any morphism in S is an S-local equivalence, i.e., a weak equivalence in LS M. Therefore
the total left derived functor
L id : Ho(M) GGA Ho(LS M)
takes the images of S to isomorphisms in Ho(LS M) (in fact, even the left derived functor L id : Ho(M) GGA LS M sends S to weak equivalences). For any other such functor
F : LS M GGA N , the only possible filler in
id
LS M
M
F
∃?
N.
is the same functor F : LS M GGA N , and it remains to prove that this is a left Quillen
functor. Since M and LS M have the same cofibrations, F : LS M GGA N preserves the
cofibrations since F : M GGA N does. To show that F preserves acyclic cofibrations, we
cannot proceed directly, but we need to show that it is part of a Quillen adjunction.
Since F : M GGA N is a left Quillen functor, it is part of a Quillen adjunction
GGA
F : M DG
G ⊥ G N : G,
which restricts into an adjunction (not Quillen yet)
GGA
F : LS M DG
G ⊥ G N : G.
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CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
Since F preserves cofibrations, this adjunction is a Quillen adjunction if and only if G
preserves fibrations. This is proved in several technical steps. First, by a general result of
Dugger (see Proposition 8.5.4 in [Hir03]), it suffices to prove that G preserves the fibrations
between fibrant objects. Second, by Proposition 3.3.16 in [Hir03], between S-local objects,
a map is a fibration in M if and only if it is a fibration in the left Bousfield localization
GGA
LS M. Finally, by Theorem 3.1.6 in [Hir03] in a Quillen adjunction F : M DG
G ⊥ G N : G and
for any class of maps S ⊆ Mor(M), the total left derived functor LF : Ho(M) GGA Ho(N )
sends the image of S to isomorphisms if and only if the right adjoint G sends fibrant objects
to S-local objects. Since it is the case, G preserves fibrations between fibrant objects, and
so preserves all fibrations. The adjunction
GGA
F : LS M DG
G ⊥G N : G
is a Quillen adjunction, and in particular, F : LS M GGA N is a left Quillen functor and
LS M satisfies the universal property of a left localization.
This section is just a brief overview of the heavy machinery of localizations of (cellular)
model categories. A complete self-contained treatment appears in [Hir03], from where we
only picked the minimum essential to be able to use it. To summarize the machinery, given
a left proper cellular model category M and a set (not a proper class) S of morphisms in
M, this machinery asserts that there is another model structure on M where
• the cofibrations are the same as in M;
• the weak equivalences are the S-local equivalences;
• the fibrations are the morphisms with the right lifting property with respect to all
cofibrations that also are S-local equivalences.
This model structure is again left proper and cellular, is denoted by LS M, and is called the
left Bousfield localization of M with respect to S. If in addition M is a simplicial model
category, then the induced structure on LS M is again a simplicial model category. The
identity functors
GGA
id : M DG
G ⊥ G LS M : id
form a Quillen adjunction, and the left Quillen functor has the property that its total left
derived functor
L id : Ho(M) GGA Ho(LS M)
sends the images of the morphisms of S to isomorphisms in the homotopy category Ho(LS M).
This is in fact the motivation of a localization of a model category : given a set (ideally
a class...) of maps in M, find another model category such that the morphisms from S
becomes isomorphisms in the new homotopy category. The left Bousfield localization is
very convenient since it explicitly constructs such a localization on the same underlying
category M. Moreover, the left Bousfield localization LS M also the universal property of
a left localization, i.e., for any other left Quillen functor F : M GGA N whose total left
derived functor LF : Ho(M) GGA Ho(N ) sends the images of S to isomorphisms, there is
a unique left Quillen functor LS M GGA N making the diagram commute
77
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
id
LS M
M
F
N,
∃! F
and this is in fact the same functor F . We will make use of this machinery more than once.
Before going on to motivic homotopy theory, we will give a criterion that we will need
later, in order to compare two (left) Bousfield localizations.
Lemma 2.3.2. Let S ⊆ T be two classes of maps in a model category M, such that the two
left-Bousfield localizations LS M and LT M exist. If every S-fibration X GGA Y between
S-fibrant objects that is a T -equivalence, also is an S-equivalence, then the two localizations
agree
LS M = LT M.
Proof. Since cof(LS M) = cof(LT M) = cof(M), it is enough to show that their weak
equivalences w. e.(LS M) = w. e.(LT M) agree. Indeed, the rest of the structure (acyclic
cofibrations, fibrations and acyclic fibrations) is forced to be the same by lifting properties.
Since S ⊆ T , it follows that T -local objects are S-local objects and therefore S-equivalences
are T -equivalences.
For the reverse inclusion, let A GGA B be a T -equivalence, and choose a fibrant replacement functor R in LS M. Consider the following commutative square
A
∼T
∼S
B
∼S
RA
RB.
Since S-equivalences are T -equivalences, the vertical fibrant replacements are T -equivalences,
∼T
and therefore so is RA GGA RB. Choose now a factorization in LS M of the map
RA GGA RB as in
∼T
A
B
∼S
∼S
∼T
RA
RB
∼S
X
∼S
where RA GA X is an S-acyclic cofibration and X GGA
A RB is an S-fibration. Since RB is
∼T
S-fibrant, then so is X. Furthermore, by the 2-out-of-3 property, the morphism X GGA RB
is a T -equivalence, and since it also is an S-fibration between S-fibrant objects, then it is an
78
CHAPTER 2. ADDITIONAL STRUCTURES ON MODEL CATEGORIES
S-equivalence by hypothesis. By the 2-out-of-3 property, RA GGA RB is an S-equivalence,
and then so is A GGA B as desired.
79
3. Motivic Homotopy Theory
Let S be a noetherian scheme of finite Krull dimension, and denote by Sm/S the category of
smooth schemes of finite type over S. The goal of this chapter is to introduce the unstable
category of motivic spaces on S, denoted by MS , which is a model category that contains
Sm/S as a subcategory. We will see to what extent this category carries homotopical informations about schemes over S, and how this is the right category for studying the homotopy
theory of schemes. In fact, this category is the first step towards cohomology theories for
schemes. As in topology, by formally inverting the suspension functor in the unstable motivic homotopy category, we obtain what is called the stable motivic homotopy category (on
S). This category is the home of various cohomology theories, i.e., objects in this category
represent such theories (a sort of Brown representability theorem). Unfortunately, I did
not have enough time to define cohomology theories for schemes, and I refer for example to
[DLØ+ 07] for the definition of these cohomology theories.
The construction of the unstable motivic category MS is done in several steps. We first
start with some category of schemes, say Sm/S, the category of smooth schemes of finite
type over a noetherian scheme S of finite Krull dimension. Except the finitness condition
which ensures that Sm/S is essentially small, all the other technical conditions are not
necessary until the very end of the construction, where they allow a useful characterization.
These categories of schemes are far from being complete and cocomplete (in the categorical
sense). The first step towards endowing them with a model structure, is to add all the
necessary limits and colimits (first axiom of a model category).
In the first section we enlarge sufficiently the category Sm/S (by formally adding all
homotopy colimits) and endow it with a first model structure. In the second section, we
follow the article [Dug01c], that treats in more generality the homotopy theory of such
cocompletions. As shows the article, despite the fact that this cocompletion is a universal
construction, it does not give an interesting homotopy theory for schemes. A problem is
that the colimits in this cocompletion does not reflect the geometry of the initial category
of schemes Sm/S. In the third section we correct these colimits by endowing the category
of schemes Sm/S with a Grothendieck topology and then localizing with respect to the
’interesting ones’. The resulting model structure is called a local model structure, and is
denoted by MS,loc . Unfortunately, the model category MS,loc does still not contain more
homotopical information than Sm/S. The article [Dug01c] shows, with two examples, that
there still is a missing ingredient : there is no object that plays a similar role as plays the
interval I in topology. The unstable homotopy category will be defined as the localization
of MS,loc with respect to the interval-like object A1 , and is the subject of study in the last
80
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
section.
3.1
Global Model Structures on Simplicial Presheaves
Let C be a small category. We will call the category of functors [C op , sSet], the category of
op
simplicial presheaves on C, and will denote it by sSetC . There are several possible model
structures on a general category of functors [C, M], where M is a model category. Among
the most well-known are the Reedy model structure, the injective and the projective model
structure. However, each of those requires some additional structure, either on C or on M.
op
In the case of sSetC , we will exploit the model structure of sSet to endow the category
op
of simplicial presheaves sSetC with the injective and the projective model structures. As
op
there will also be local (injective and projective) model structures on sSetC , the structures
from this section will usually be called the global injective model structure and the global
projective model structure.
For the time being, let’s work in a more general setting. Given a small category C and
a model category M, there are two ’natural’ ways to endow the category of diagrams MC
with a model structure, the injective model structure and the projective model structure.
The global injective model structure is the one with
• cofibrations : objectwise cofibrations;
• weak equivalences : objectwise weak equivalences;
• fibrations : maps satisfying the right lifting property with respect to trivial cofibrations;
while the global projective model structure has
• fibrations : objectwise fibrations;
• weak equivalences : objectwise weak equivalences;
• cofibrations : maps satisfying the left lifting property with respect to trivial fibrations.
However, these two model structures need not exist for an arbitrary model category M.
Indeed, the injective model exists whenever M is combinatorial1 , and for the projective
model we only need M to be cofibrantly generated. The additional structure of the model
category M is also inherited by the category of diagrams MC . We will focus on the projective
model structure and refer to Proposition A.2.8.2 in [Lur09] for the proof of the existence of
the injective model structure.
3.1.1
The global projective model structure
Let C disc denote the discrete category associated to C, i.e., the category with the same objects
disc
as C and only identities as morphisms. The idea of the proof is to first endow MC
with
disc
C
C
GGA
a ’discrete’ model structure, find an adjoint pair of functors M
DG
G ⊥ G M and lift the
disc
C
C
model structure of M
to M with Kan’s theorem 2.1.9.
1
Recall that a model category is called combinatorial if the model structure is cofibrantly generated and
the underlying category is locally presentable, see definition 2.1.6.
81
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
The cofibrantly generated model structure on MC
proposition.
disc
is a particular case of the following
Proposition 3.1.1. Let S be a set, and for any element s ∈ S, let Ms be a cofibrantly
generated model category with generating cofibrations Is and generating trivial cofibrations
Q
Js . There is a cofibrantly generated model structure on the product category s MS in which
a map is a cofibration, fibration or weak equivalence if and only if it is in each component.
Proof. For any s ∈ S, denote the identity map between the initial objects by 1∅,s : ∅ GGA
∅ ∈ Ms , and consider the sets

I=
[
s∈S
Is ×


Y
{1∅,t }
J=
and
[
Js ×
s∈S
t6=s

Y
{1∅,t } .
t6=s
A map in s Ms is a fibration if and only if it has the right lifting property with respect to
any map in J, i.e., if every component has the lifting property with respect to any map in
Is , i.e., if every component is indeed a fibration. Similarly for trivial fibrations, cofibrations
and trivial cofibrations.
Q
Corollary 3.1.2. Let C be a small category and M be a cofibrantly generated model category.
disc
Then the category of diagrams MC
is a cofibrantly generated model category where a
map is a cofibration, fibration or weak equivalence if and only if it is in each component
respectively a cofibration, fibration or weak equivalence.
Proof. This is the last proposition where the set S is the set of objects Ob(C).
disc
C
∗
GGA
Let’s now define the adjunction F : MC
DG
G ⊥ G M : i that will allow the cofibrantly
disc
C
C
generated model structure of M
to be ’lifted’ up to M . Denote by i : C disc ,GGA C the
embedding of categories, and by
i∗ : MC GGA MC
disc
the functor induced by precomposition with i. This functor admits a left Kan extension
disc
operation, which is its left adjoint denoted by F , defined on objects S ∈ MC
by
a
F (S) =
a
S(C),
C∈C C(C,−)
i.e., it sends an object D ∈ C to
a
a
s
S(C) ∈ M. On a morphism S GGA S 0 ∈ MC
disc
,
C∈C C(C,D)
it is defined by the morphisms induced by
s(C)
S(C) GGA S 0 (C) GGA
a
S 0 (C),
C(C,D)
for every objects C, D ∈ C and every arrow C → D. See [Hov99, 11.5.6] for the motivation
of this definition.
Lemma 3.1.3. The pair of functors F : MC
disc
82
C
∗
GGA
DG
G ⊥ G M : i is an adjoint pair of functors.
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Proof. Given S ∈ MC
disc
and T ∈ MC , we want to show that there is a natural bijection
MC (F (S), T ) ∼
= MC
disc
(S, i∗ (T )).
The left-hand side may be rewritten as

MC (F (S), T ) = MC 

a

S(C), T  ∼
=
a
C∈C C(C,−)
Y

MC 
C∈C
a
S(C), T ,
C(C,−)
and the right-hand side as
MC
disc
(S, i∗ (T )) ∼
=
Y
M(S(C), T (C)),
C∈C
since the discrete category C disc has only identities as morphisms. Using the notation X =
S(C), define a function


φ : MC 
X, T  GGA M(X, T (C))
a
C(C,−)
that sends a morphism
`
C(C,−) X
g
GGA T to the composite
X GGA
gC
X GGA T (C)
a
C(C,C)
where X is embedded into the copy of X associated to the identity map C GGA C.
f
To see that φ is onto, fix a morphism X GGA T (C) ∈ M. We need to produce a natural
`
`
transformation XC(C,−) =⇒ T , i.e., compatible morphisms C(C,D) X GGA T (D), i.e.,
which for any s ∈ C(C, D) are given by the composite
T (s) ◦ f
X
T (D).
f
T (s)
T (C)
To see that φ is a one-to-one function, suppose that there are two natural transformations
`
f, g : C(C,−) X =⇒ T such that φ(f ) = φ(g). For a fixed D ∈ C and a morphism s ∈
C(C, D), consider the following diagram
gC
a
X
T (C)
fC
C(C,C)
iid
s∗
T (s)
X
jD,s
is
a
gD
X
T (D).
fD
C(C,D)
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Since gC iid = fC iid by hypothesis, and since T (s)gC = gD s∗ and T (s)fC = fD s∗ because
they are natural transformations, there is a map gD is = jD,s = fD is for any s ∈ C(C, D).
Since for any such s they are equal to the upper part of the diagram, they give a compatible
`
cocone, and so there is a unique filler fD = gD : C(C,D) X GGA T (D). Doing the argument
for any object D ∈ C shows that f = g as desired.
This is the adjunction that will lift the cofibrantly generated model structure from MC
to MC . Denote by


[
IC disc =
Y
I ×
C∈C
disc
{1∅,D }
D6=C
disc
the generating set of cofibrations of MC , where I is the generating set of cofibrations of
M, and similarly by JC disc the generating set of trivial cofibrations. We need now a method
to show that the two sets F (IC disc ) and F (JC disc ) in MC , permit the small object argument.
We need two lemmas, one of which was already proved in lemma 3.1.3.
Lemma 3.1.4. Let M be a cocomplete category, C be a small category, and let C be an
object in C. There is an adjoint pair of functors
C
GGA
F C : M DG
G ⊥ G M : evC
where F C (X) =
`
C(C,−) X
and evC (T ) = T (C).
C disc : ev .
GGA
Proof. Compose the adjunction of lemma 3.1.3 with the adjunction iC : M DG
C
G ⊥G M
Lemma 3.1.5. Let M be a cocomplete category, C a small category and C an object in C.
Pick a map A GGA B ∈ M. For every pushout square
a
a
A
C(C,−)
B
C(C,−)
p
S
T
in MC and any object D ∈ C, there is an objectwise pushout square
a
a
A
C(C,D)
B
C(C,D)
p
S(D)
T (D)
84
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
in M.
Proof. This follows from the fact that colimits (and in particular pushouts) can be computed
objectwise in any functor category.
Theorem 3.1.6 (Model structure on the category MC of C-diagrams). Let C be a small
category and M a cofibrantly generated model category with generating set of cofibrations I
and generating set of trivial cofibrations J. Then there exists a cofibrantly generated model
structure on the category of C-diagrams MC with generating set of cofibrations F (IC disc ),
generating set of trivial cofibrations F (JC disc ) and where
• fibrations : objectwise fibrations;
• weak equivalences : objectwise weak equivalences;
• cofibrations : retract of transfinite compositions of pushouts of maps from F (IC disc ).
disc
C
∗
GGA
Proof. Consider the adjunction F : MC
DG
G ⊥ G M : i of lemma 3.1.3, and denote by
disc
F (IC disc ) and F (JC disc ) the generating sets of cofibrations, resp. trivial cofibrations in MC
g
defined in corollary 3.1.2 (everything is objectwise). For any map A GGA B ∈ M,


F g ×
Y

`
a
1∅,D  = 
A GGA
B  ∈ MC ,
a
C(C,−)
C(C,−)
D6=C

g
where 1∅,D : ∅ GGA ∅, and C is the component associated to the map g in the identification
Q
Q
disc
∼
MC
= C∈C M. Recall that the maps of the form g × D6=C 1∅,D , where g ∈ I, are the
disc
generating cofibrations for MC , so that
F (IC disc ) =

 a
A GGA

C(C,−)
a
B
C(C,−)



,
A→B∈I
C∈C
and we obtain the set of generating trivial cofibrations F (JC disc ) similarly.
To show that they permit the small object argument, treating the case of cofibrations
`
for example, we need all C(C,−) A to be small relative to F (IC disc ), where A GGA B ∈ I.
`
So let’s fix an element C(C,−) A that arises as the domain of a map in F (IC disc ). Given a
big enough cardinal λ and an appropriate λ-sequence Xλ , there are the following natural
isomorphisms
colimλ MC (
a
3.1.4
∼
=
A, Xλ )
s.o.a.
∼
=
∼
=
3.1.4
∼
=
where the coproducts are more precisely
ment’. In fact, the natural map
`
colimλ M(A, Xλ (C))
M(A, colimλ (Xλ (C)))
M(A, (colimλ Xλ )(C))
MC (
C(C,−)
a
A, colimλ Xλ ),
and s.o.a. means the ’small object argu-
colimλ M(A, Xλ (C)) GGA M(A, colimλ (Xλ (C)))
85
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
is an isomorphism for big enough λ, since the λ-sequence X∗ (C) is indeed a transfinite
composition of pushouts of maps from I, by lemma 3.1.5.
In order to apply Theorem 2.1.9, the second condition to be checked is that the functor
∗
i takes relative F (JC disc )-cell complexes to weak equivalences. Recall that
F (JC disc ) =

 a

A GGA
C(C,−)
a
B



C(C,−)
.
A→B∈J
C∈C
By lemma 3.1.5, a relative F (JC disc )-cell complex is objectwise a relative J-cell complex,
disc
which is in particular an objectwise weak equivalence, i.e., a weak equivalence in MC .
Under these hypothesis, Kan’s theorem 2.1.9 applies and gives a cofibrantly genereated
model structure on MC . Moreover, using the adjunction
MC (
a
A, T ) ∼
= M(A, T (C)),
C(C,−)
a diagram
`
`
has a filler
`
C(C,−) B
C(C,−) A
S
C(C,−) B
T
=⇒ S if and only if the induced diagram
A
S(C)
B
T (C)
has a filler A GGA S(C). Applying this to all maps in F (JC disc ) implies that fibrations
in MC , which are the maps that have the right lifting property with respect to any map
in F (JC disc ), are exactly the objectwise fibrations, which are the maps that have the right
lifting property with respect to J. Kan’s theorem also says that the weak equivalences are
the morphisms that i∗ takes to weak equivalences, i.e., they are exactly the objectwise weak
equivalences.
Applying this theorem to the cofibrantly generated model category M = sSet, where
the generating sets of cofibrations and trivial cofibrations are
{∂∆[n] GGA ∆[n]}n∈N
and
or more precisely given by
86
n
o
Λk [n] GGA ∆[n]
k≤n ,
0<n
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
• cofibrations : monomorphisms, i.e., degree-wise injective maps;
• weak equivalences : weak homotopy equivalences, i.e., inducing isomorphisms on all
homotopy groups of the geometric realization (for any choice of basepoint);
• fibrations : Kan fibrations, i.e., satisfying the right lifting property with respect to the
maps ∂∆[n] ,GGA ∆[n];
op
gives the desired projective model structure on simplicial presheaves sSetC .
op
Corollary 3.1.7 (Global projective model structure on sSetC ). Let C be a small category.
op
There is a cofibrantly generated model structure on sSetC , called the global projective model
structure, with generating set of cofibrations
I=

 a
∂∆[n] GGA

C(C,−)
a


,
∆[n]

C(C,−)
n∈N
C∈C
and generating set of trivial cofibrations
J=

 a

Λk [n] GGA
a
C(C,−)
C(C,−)


.
∆[n]

k≤n,0<n
C∈C
Moreover, the fibrations (resp. weak equivalences) are exactly the objectwise fibrations (resp.
weak equivalences).
op
There is a similar theorem about an injective model structure on sSetC , which becomes
in fact a combinatorial model structure, see [Lur09]. In the injective model, the cofibrations
and weak equivalences are the objectwise ones, and the fibrations are defined by a lifting
property.
Moreover, the injective and projective model structures on MC are left/right proper if
the model structure on M is left/right proper.
Proposition 3.1.8. Let M be a cofibrantly generated model category and let C be a small
category. If M is left (resp. right) proper, then so are the injective and projective model
structures on MC .
Proof. The colimits and the limits can be computed object-wise in the functor category
MC . In particular, a pushout or pullback in MC induces objectwise pushouts and pullbacks
in M. Since in both models, cofibrations and fibrations in MC are in particular objectwise cofibrations and fibrations, we are in the right setting to apply the properness of M.
The results follows since in both models the weak equivalences are exactly the object-wise
ones.
Similarly, these two model structures are simplicial if the initial model structure on M
is simplicial. We will prove it for the projective model structure.
Proposition 3.1.9. Let M be a cofibrantly generated model category and let C be a small
category. If the model structure on M is simplicial, then so is the projective model structures
on MC .
87
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Proof. We have already seen that MC is tensored ans cotensored as enriched over sSet,
where for M ∈ M and C ∈ C the tensors and cotensors are
M ⊗C: D G
[ GA M (D) ⊗ C ∈ M
and
MC : D G
[ GA M (D)C ∈ M.
Moreover, since the fibrations in MC are in particular object-wise fibrations (of simplicial
sets), for any cofibration K• ,GGA L• ∈ sSet and any fibration X GGA
A Y ∈ M, the induced
map
X L GGA
A X K ×Y K Y L
∈M
is an object-wise fibration. Moreover since the object-wise fibrations are exactly the fibrations in MC , the this morphism is a fibration in MC . By the same argument, if either i or
p is a weak equivalence, then so is this induced morphism.
All in all, using the fact that the model structure on sSet is also cellular (since in fact
it is locally finitely presentable), we have the following result.
Corollary 3.1.10. Let C be a small category. There are two cellular, proper, simplicial
op
model structures on sSetC , called the global projective model structure and the global injective model structure sharing the same weak equivalences, which are the object-wise weak
equivalences.
The fibrations in the projective model are exactly the object-wise fibrations and the cofibrations in the injective model are exactly the object-wise cofibrations. The rest of the structure is (uniquely) determined by intersections and lifting properties.
3.1.2
Comparison between injective and projective global models
Since these two models have the same weak equivalences and the same underlying category,
they both are models for the same homotopy category. Moreover, a fibration in the injective
model (resp. a cofibration in the projective model) is also an objectwise fibration (resp.
cofibration), since a global lifting is in particular an objectwise lifting. In particular, the
projective model has fewer cofibrations, and the injective model has fewer fibrations. We
could therefore hope for a Quillen equivalence between those model structures.
Proposition 3.1.11 (Quillen equivalence between the global injective and the global projective models). The identity functors
op
C op
GGA
id : sSetCproj DG
G ⊥ G sSetinj : id
form a Quillen equivalence.
Proof. By the remark above, these two functors form a Quillen pair, which is indeed a
Quillen equivalence, since the two categories have the same weak equivalences.
Each model has its own cofibrant and fibrant objects. The following proposition gives
what characterizations can be given for the fibrant and cofibrant objects.
Proposition 3.1.12 (Cofibrant and Fibrant objects in the global model structures). In the
op
global injective model structure of sSetCinj
88
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
• all the objects are cofibrant;
• there is no good characterization for the fibrant objects.
op
In the global projective model structure of sSetCproj
• the representables rC = C(−, C) are still cofibrant;
• the fibrant objects are the ones that are objectwise (Kan) fibrant simplicial sets.
Proof. Clearly all the objects are cofibrant in the injective model, since the cofibrations are
the monomorphisms, i.e., objectwise injective.
A representable C(−, C) is cofibrant in the projective model if and only if the diagrams
F
∃?
C(−, C)
G
op
admit liftings for all trivial fibrations F GGA G ∈ sSetCproj . Since trivial fibrations in the
projective model are exactly the objectwise trivial fibrations, and since C(−, C) is a discrete
simplicial presheaf, this is equivalent as requiring liftings in all diagrams
F (−)n
∃?
C(−, C)
G(−)n
of presheaves on C. However, all these liftings exists by the Yoneda lemma, and so C(−, C)
is cofibrant. Moreover, since fibrations are objectwise fibrations in the projective model,
op
F ∈ sSetCproj is fibrant if and only if each F (C) is a fibrant simplicial set.
In addition to these two structures, there is another global intermediate model structure
op
on sSetC , the flasque model structure. Intermediate means here that the identity functors
are part of Quillen equivalences
op
C op
C op
GGA
GGA
sSetCproj DG
G ⊥ G sSetflasque DG
G ⊥ G sSetinj .
A description of the flasque model structure is given by Isaksen in [Isa05].
Let’s also mention that if the indexing category C has a Reedy structure, the category
of C-diagrams MC endowed with the Reedy model structure, is also an intermediate model
structure, again in the sense that there are Quillen equivalences
op
C op
C op
GGA
GGA
sSetCproj DG
G ⊥ G sSetreedy DG
G ⊥ G sSetinj .
89
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
3.2
Universal Model Categories and Small Presentations
This section summarizes the article [Dug01c]. Given a small category C, it is possible
to embed it into a model category, in a universal way. The resulting model category,
denoted U C, is called the universal model category built from C and is obtained by formally
(freely) adding all homotopy colimits and then endowing it with an appropriate model
structure. The localization process may come into play afterwards to impose relations on
this universal model category U C, for several potential reasons. One possible reason is
to recover some structure from C that was lost by freely expanding it into U C, since the
embedding C ,GGA U C does not preserves colimits in general.
A localization of U C is very similar to presentations in algebra. The category C may be
seen as the set of generators, and the localization process quotients out by the relations one
wishes to impose. The methods explained in this section are used to give a presentation of
our category of interest, some category of motivic spaces, Quillen equivalent with the model
category that was originally defined in [MV99].
3.2.1
Universal cocompletion
Let’s fix a small category C. The construction of the universal model category U C mimics
a general construction in category theory, by which one formally adds colimits to C by
op
embedding it in its category of presheaves C ,GGA SetC 2 . The canonical functor r : C ,GGA
op
SetC is called the Yoneda embedding, and sends any object C ∈ C to the representable
op
presheaf rC ∈ SetC defined as
rC = C(−, C) : C op GGA Set : D G
[ GA C(D, C).
As in any category of functors, the colimits in [C op , Set] can be computed object-wise. This
means that for any small category I and functor F : I GGA [C op , Set], its colimit exists3
and is given by
colim F : C op GGA Set : C G
[ GA colimi∈I F (i)(C)
and with the natural transformation F =⇒ ∆ colim F coming from the colimits in Set.
Indeed, the goal was to construct a cocomplete category, which is universal in some sense.
op
Note that it happens in fact that SetC is also complete, since Set is. This free cocompletion
construction is universal among cocompletions of C, as stated in the following proposition.
Proposition 3.2.1 (Universal Property of Presheaves). Any functor γ : C GGA D where D
is cocomplete can be factored uniquely up to unique isomorphism through a colimit-preserving
op
functor Re : SetC GGA D, as in the diagram
r
C
SetC
γ
op
Re
D.
2
3
See Discussion 2.2.6 in [Dug] for some intuition about this construction.
because it exists object-wise in Set.
90
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Moreover, the functor Re is part of an adjoint pair of functors Re : SetC
op
GGA
DG
G ⊥ G D : Sing.
Proof. The construction of the functor Re relies on the fact that any presheaf can be canop
nonically written as a colimit of representable presheaves. Indeed, letZX ∈ SetC be any
presheaf. The Grothendieck construction on the functor X, denoted by
X, is the category
with
• Objects : pairs (C, x) where x ∈ X(C);
f
• Morphisms : (C, x) GGA (C 0 , x0 ) is a C-morphism C GGA C 0 such that X(f )(c0 ) = c.
There is a projection functor that fits into the picture with the Yoneda embedding
Z
X
π
GGA
C
r
GGA
SetC
op
(C, x) G
[ GA C G
[ GA rC = C(−, C).
R
∼ colim
◦
The claim is now that X =
X r π. First, by the Yoneda lemma, any element
x ∈ X(C) represents and is represented by a unique morphism
r ◦ π(C, x) = C(−, C) =⇒ X.
This shows that X is a cocone for the functor r ◦ π. Given any other cocone Y , we need to
find a morphism T : X GGA Y such that every triangle
r ◦ π(C, x)
X
TC
Y
commutes. By the Yoneda lemma, every structure map of Y , part of the cocone-data, which
is of the form
r ◦ π(C, x) GGA Y,
corresponds to a unique element yx ∈ Y (C). Again by the Yoneda lemma, the desired
morphism exists and is moreover necessarily defined object-wise by the set-theoretic formula
TC : X(C) GGA Y (C) : x G
[ GA yx .
This proves that X is a universal cocone, and hence naturally isomorphic to the colimit
X∼
= colim r ◦ π.
op
The functor Re : SetC GGA D can then be defined on representables, and extended to
all presheaves by being forced to preserve all colimits, i.e., it keeps in mind how the presheaf
is constructed as a colimit of representables and builds it again in D as
Re X ∼
= colimR
X
r ◦ π := colimR
op
For any presheaf X ∈ SetC , the diagram is the following
91
X
γ ◦ π.
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Z
X
π
r
C
SetC
γ
op
Re
D.
As for the representables, they are necessarily defined by Re(C(−, C)) := γ(C). Note
that the functor Re preserves colimits by construction. Any other such (colimit-preserving)
functor is forced to agree with Re on representables, and therefore is canonically isomorphic
to Re by the naturality of the colimit.
The right adjoint to Re is the singular functor that is defined by
op
Sing : D GGA SetC : D G
[ GA SingD,
where SingD is the presheaf given by the formula C G
[ GA D(γ(C), D). To see that Re and
op
Sing really are adjoints, note that for X ∈ SetC and any object D ∈ D, we have the
natural isomorphisms
D(Re(X), D)
∼
=
∼
=
D(colim γ ◦ π(C, x), D)
∼
=
lim SingD(C)
(?)
lim D(γ ◦ π(C, x), D) = lim D(γ(C), D)
op
op
∼
=
∼
=
lim SetC (C(−, C), SingD) = lim SetC (r ◦ π(C, x), SingD)
∼
=
SetC (X, SingD),
op
SetC (colim r ◦ π(C, x), SingD)
op
where (?) is a natural isomorphism by theZ Yoneda lemma, and all limits and colimits are
X.
taken over the Grothendieck construction
3.2.2
Universal homotopy cocompletion
We will now generalize this situation homotopically. As the category of simplicial sets sSet
plays a similar role in homotopy theory as the category Set does in general category theory,
op
we will add a simplicial dimension to the category SetC by embedding it in the category
op
of simplicial presheaves sSetC . There is a canonical embedding sending any presheaf to a
constant (discrete) simplicial presheaf
F : C op GGA Set
G
[ GA
F• : C op GGA sSet,
which is defined by Fn (C) = F (C) in all dimensions n ∈ N, and with only identities as faces
and degeneracies. By abuse of language, the simplicial presheaf F∗ will often be denoted by
F , and the composite of embeddings
r
C ,GGA SetC
op
,GGA sSetC
op
will also be denoted by r, when there is no confusion about whether rX is simplicial or not.
op
We would now like to endow the category of diagrams sSetC with a model structure that
92
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
turns it into a universal model category satisfying a universal property similar to that of
op
SetC . When working with model categories, a filler in a diagram
r
C
sSetC
op
γ
M.
should be a map of model categories, i.e., half of a Quillen pair. Let’s fix the following
convention : a Quillen pair (or Quillen adjunction)
GGA
L : M DG
G ⊥G N : R
will be written as a map of model categories M GGA N , i.e., we pick the left Quillen
functor of the adjunction. The uniqueness ’up to unique isomorphism’ condition may also
be rephrased in a homotopy setting, i.e., some sort of uniqueness up to homotopy. This is
done by noticing that in the case of general category theory, the uniqueness up to unique
isomorphism means that some category of choices is a contractible4 groupoid. In the example
op
op
of proposition 3.2.1, given a functor r : C GGA sSetC into a cocomplete category sSetC ,
the category of choices would be
• Objects : colimit-preserving functors Re : sSetC
op
GGA D;
• Morphisms : natural transformations Re =⇒ Re0 .
In a homotopy setting, the maps of interest are more often just weak equivalences rather
than isomorphisms. The category of choices will therefore rarely be a groupoid, and the key
property of ’homotopically universal’ constructions is that the category of choices is still
contractible. Moreover, asking for strictly commuting diagrams is often too restrictive, we
can only ask the diagrams to commute up to natural transformations5 .
In our case of interest, we want a universal model category C GGA U C such that for any
functor C GGA M to a model category, there is a map of model categories ’unique up to
homotopy’, such that the triangle
C
r
sSetC
op
γ
M
commutes. In this case, the category of factorizations is given by
4
A small category is said to be contractible if the realization of its nerve is a contractible topological
space.
5
See the paragraph after proposition 3.2.2 for the reason why we require the natural transformation in
our situation.
93
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
• Objects : triples (L, R, η) such that
L : sSetC
op
GGA
DG
G ⊥G M: R
is a Quillen pair and η : L ◦ r =⇒ γ is a natural transformation;
• Morphisms : (L, R, η) GGA (L0 , R0 , η 0 ) are natural transformations L =⇒ L0 such that
all triangles commute
L0 (rX)
L(rX)
ηrX
0
ηrX
γ(X).
op
Let’s now define a model structure on our category of simplicial presheaves sSetC . Between
the two model structures defined in the last section, the projective and the injective, the
projective model is more appealing since it has fewer cofibrations, and therefore the chances
of having a left Quillen map out of it are greater. Recall that the projective model structure
is defined by setting a map T : X =⇒ Y to be a fibration (resp. weak equivalence) if
TC : X(C) GGA Y (C) is a fibration (resp. weak equivalence) for all objects C ∈ C, and the
cofibrations are defined to be the morphisms with the left lifting property with respect to
the trivial fibrations. The category of simplicial presheaves on C with the projective model
structure will sometimes be denoted by U C, referring to the fact that it is the universal
model category built from C in the following sense.
Proposition 3.2.2 (Universal Property of Simplicial Presheaves). Any functor γ : C GGA
M into a model category M can be ’factored through’ U C in the sense that there is a Quillen
∼
GGA
adjunction Re : U C DG
G ⊥ G M : Sing and a natural pointwise weak equivalence η : Re ◦ r =⇒ γ
as in the diagram
r
C
UC
=⇒
γ
Re
M.
Moreover, the category of such factorizations (as defined above) is contractible.
By first ignoring the model structures, proposition 3.2.1 gives us a colimit-preserving
op
functor Re : SetC GGA M
C
SetC
op
U C = sSetC
Re
γ
M,
94
op
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
that we want to further extend to a left-adjoint functor Re : U C GGA M. By adding
op
op
a simplicial direction from SetC to sSetC , we are now faced with understanding the
realization of simplicial objects of the form rX ⊗ ∆[n] ∈ U C. Indeed, proposition 3.2.2
will be proved by making a correspondence with the theory of cosimplicial resolutions, from
which we first recall some definitions.
Denote by cM = [∆, M] = M∆ the category of cosimplicial objects of M, which is
endowed with the Reedy model structure6 . Write also c∗ : M ,GGA M∆ for the embedding X G
[ GA c∗ X, where c∗ X is the constant diagram concentrated in X. A cosimplicial
resolution of an object X ∈ M is
• a Reedy cofibrant object ΓX ∈ cM, with
∼
• a (pointwise) weak equivalence ΓX GGA c∗ X ∈ cM.
Let now C be a small category. A cosimplicial resolution of a functor γ : C GGA M is
• a functor Γ : C GGA cM such that each Γ(C) is Reedy cofibrant for all C ∈ C, with
∼
∼
• a natural transformation Γ =⇒ c∗ ◦ γ that is a (pointwise) weak equivalence Γ(C) GGA
c∗ ◦ γ(C).
This can be seen as a sort of ’functorial, cofibrant, lifting up to homotopy of γ along c∗ ’, as
in
C
Γ
cM
c∗
γ
M.
A map between two such cosimplicial resolutions Γ1 and Γ2 is a natural transformation
Γ1 =⇒ Γ2 such that all the diagrams
Γ1 (C)
Γ2 (C)
c∗ ◦ γ(C)
commute. The category of cosimplicial resolutions of γ together with this notion of maps
between them will be denoted by coRes(γ). Given our problem of finding a filler to
C
r
UC
∃?
γ
M,
6
The only important thing is that the the weak equivalences are exactly the objectwise ones. See Chapter
15 in [Hir03] for more details.
95
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
define the category Factγ (U C) of factorizations of γ through U C to be
GGA
• Objects : Quillen adjunctions Re : U C DG
G ⊥ G M : Sing together with a natural transfor∼
mation η : Re ◦ r =⇒ γ that is a (pointwise) weak equivalence;
• Morphisms : A morphism (Re, Sing, η) GGA (Re0 , Sing0 , η 0 ) is a natural transformation
Re =⇒ Re0 such that the diagrams commute
Re0 (rC)
Re(rC)
ηC
η0C
γ(C).
Proof of proposition 3.2.2. The sketch of the proof follows the proof of Proposition 2.3 in
[Dug01c]. We want to show that this category Factγ (U C) of factorizations of γ ’through’ U C
is not empty and is moreover contractible (second claim). Indeed, the category coRes(γ) of
cosimplicial resolutions of γ admits these properties, see for example Chapter 16 in [Hir03],
GGA
so it would be enough to show that there is an equivalence of categories coRes(γ) DG
G ⊥G
Factγ (U C). Intuitively, both steps
γ ∈ MC
Γ ∈ [∆, MC ]
and
Re ∈ [∆op , MPre(C) ] ∼
= [U C, M],
Re ∈ MPre(C)
are obtained by adding a simplicial direction to the initial functors γ and Re.
Given objects S ∈ Set and M ∈ M, there exists a ’tensor product’7 defined by M ⊗S :=
`
S M , the coproduct of copies of M . The objects in this product can be simplicially enriched
to give a ’tensor product’ in M between K• ∈ sSet and X ∗ ∈ cM, defined by the formula

X ∗ ⊗ K• = coeq 

a
X k ⊗ Km ⇒
[k]→[m]
a
X n ⊗ Kn  ∈ M,
[n]
X(α)⊗id
where for a given α : [k] → [m], the two morphisms are induced by X k ⊗Km GGA X m ⊗Km
id ⊗K(α)
and by X k ⊗ Km GGA X k ⊗ Kk . As the first product satisfies the adjoint relation
M(M ⊗ S, M 0 ) ∼
= M(
a
S
M, M 0 ) ∼
=
Y
M(M, M 0 ) ∼
= Set(S, M(M, M 0 )),
S
this tensor product satisfies a similar adjoint relation
M(X ∗ ⊗ K• , M ) ∼
= sSet(K• , M(X ∗ , M )),
where M(X ∗ , M ) is the simplicial set whose n-simplices are the set M(X n , M ). The objects
in this product may be further enriched to give a ’tensor product’ in M of C-diagrams.
7
This is the tensor of S and M when seeing M as enriched over Set.
96
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Indeed, given a functor Γ : C GGA cM and a simplicial presheaf F : C op GGA sSet, the
formula


a
Γ ⊗C F = coeq 
Γ(a) ⊗ F (b) ⇒
a
Γ(c) ⊗ F (c) ∈ M,
c∈C
a→b∈C
defines an object in M and satisfies the adjoint relation
op
M(Γ ⊗C F, M ) ∼
= sSetC (F, M
:: (Γ, M )),
where similarly M
[ GA M(Γ(c)∗ , M ).
:: (Γ, M ) is defined to be the simplicial presheaf c G
Let’s now give the correspondence between coRes(γ) and Factγ (U C). Starting with a
∼
cosimplicial resolution Γ : C GGA cM and a natural weak equivalence η : Γ =⇒ c∗ ◦ γ, first
define the realization by the formula
Re : U C GGA M : F G
[ GA Re(F ) = Γ ⊗C F.
The singular functor is given by
Sing : M GGA U C : M G
[ GA [c 7→ M(Γ(c)∗ , M )] ,
GGA
and the pair Re : U C DG
G ⊥ G M : Sing is an adjoint pair of functors by the above adjunction.
∼
The natural weak equivalence Re ◦ r =⇒ γ will be exactly η, since Re ◦ r(c) ∼
= Γ(c)0 . Indeed
when F = r(c) = C(−, c) is a representable (constant) simplicial presheaf, the formula
becomes


Γ ⊗C F = coeq 
GGA
Γ(a) ⊗ C(b, c) GGA
a
a→b∈C
a
Γ(d) ⊗ C(d, c) ,
d∈C
where C(b, c) and C(d, c) are now constant presheaves. This fact allows the computation of
the internal terms
Γ(a) ⊗ C(b, c) ∼
= coeq
"
GGA
Γ(a)k ⊗ C(b, c) GGA
a
k→m
∼
= coeq
"
GGA
Γ(a) GGA
a
k
k→m
#
a
Γ(a)n ⊗ C(b, c)
n
#
a
Γ(a)
n
⊗ C(b, c)
n
∼
= Γ(a)0 ⊗ C(b, c)
a
=
Γ(a)0 ,
C(b,c)
where − ⊗ C(b, c) commutes with colimits because it is a left adjoint, and since in the
coequalizer everything gets quotiented out, only the copy of Γ(a)0 associated to the identity
id
morphism 0 GGA 0 ∈ ∆ remains. Also, the last equality holds because the tensor product
has been reduced to the cotensor of an object of M with a set, which is a coproduct.
Plugging into the initial product gives

Γ ⊗C F = coeq 


a
a→b∈C
a

GGA
Γ(a)0  GGA
C(b,c)


d∈C
97

a
a
C(d,c)
Γ(d)0  ,
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
where the two morphisms are induced by Γ(a)0 GGA Γ(b)0 GGA
`
0
C(b,c) Γ(b)
and by
α
Γ(a)0 GGA C(a,c) Γ(a)0 . For every morphism a GGA b ∈ C, and a chosen f ∈ C(b, c),
the coequalizer diagram
`
0
C(b,c) Γ(b)
`
α∗
Γ(a)0
f∗
`
C(b,c) Γ(a)
0
`
d∈C
`
C(d,c) Γ(d)
0
id∗
0
C(a,c) Γ(a)
`
says that the copy of Γ(a)0 associated to the morphism f ∈ C(b, c) is identified with a copy
of Γ(b)0 . Similarly, this copy of Γ(b)0 gets itself identified with a copy of Γ(c)0 , which gets
id
itself identified with the copy Γ(c)0 corresponding to the identity morphism c GGA c of
C(c, c). All in all, the coequalizer is

Γ ⊗C F = coeq 

a

a

a→b∈C
GGA
Γ(a)0  GGA
C(b,c)

a

a

d∈C
∼ Γ(c)0 ,
Γ(d)0  =
C(d,c)
∼
and so the natural weak equivalence Re ◦ r =⇒ γ is the same as the 0th position of the given
natural weak equivalence Γ =⇒ c∗ ◦ γ. Before giving the other direction, it remains to prove
that the adjoint pair
GGA
Re : U C DG
G ⊥ G M : Sing
is in addition a Quillen pair. This can be done by checking that Sing sends (trivial) fibrations
in M to (trivial) fibrations in U C, so let M GGA M 0 ∈ M be a (trivial) fibration. Its image
in U C is the induced C-diagram
M(Γ(−)∗ , M ) =⇒ M(Γ(−)∗ , M 0 ),
which is indeed a trivial (fibration) in the projective structure of U C, if and only if it is an
objectwise (trivial) fibration, i.e., that
M(Γ(C)∗ , M ) =⇒ M(Γ(C)∗ , M 0 ),
is a (trivial) fibration. This is Corollary 5.4.4 in [Hov99], which is proved with the language
of framings.
GGA
For the other direction, suppose we are given a Quillen pair Re : U C DG
G ⊥ G M : Sing with
∼
◦
a natural weak equivalence η : Re r =⇒ γ. For any object C ∈ C, we get a cosimplicial
resolution of γ(C) by defining
Γ(C) : n G
[ GA Re(rC ⊗ ∆[n]).
op
The product rC ⊗ ∆[n] is the product of the monoidal category sSetC , where ∆[n] is seen
as a constant simplicial presheaf. This is just the product of sSet extended objectwise to
98
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
the category of C-diagrams8 . The cosimplicial resolutions Γ(C) are cofibrant since both rC
(a representable) and ∆[n] are. Moreover, the weak equivalence Γ =⇒ c∗ ◦ γ is given by the
composite of weak equivalences
Re(rC)
∼
∼
Γ(C)n
γ(C).
GGA
This correspondence gives a natural equivalence of categories coRes(γ) DG
G ⊥ G Factγ (U C).
The fact that coRes(γ) is not empty is Proposition 16.1.9 in [Hir03], while the fact that it
is contractible is Theorem 16.7.5.
Note that we could not hope for the triangles to commute without the help of a natural
transformation. Indeed, by applying the Yoneda lemma in each dimension, any map ∅ GGA
rC = C(−, C) has the left lifting property with respect to trivial fibrations and hence any
representable rC is cofibrant in the model structure U C. Furthermore, since Re preserves
cofibrations and colimits, all the objects Re(rC) are still cofibrant in M, a property that
γ(C) might not necessarily share.
3.2.3
Small presentations
GGA
For some model categories M, the Quillen adjunction Re : U C DG
G ⊥ G M : Sing can be turned
into a Quillen equivalence by localizing U C at some class of maps. This procedure gives the
category M what is called a small presentation.
Definition (Small Presentation). A small presentation of a model category M consists of
(1) a small category C (the generators);
GGA
(2) a Quillen adjunction Re : U C DG
G ⊥ G M : Sing;
(3) a set S of maps in U C (the relations);
and we require the properties that
(a) the left derived functor of Re turns all the maps of S into weak equivalences in M;
GGA
(b) the induced Quillen adjunction Re : U C/S DG
G ⊥ G M : Sing is indeed a Quillen equivalence.
A small presentation may be thought of as giving generators and relations as in algebra.
The small category C represents the generators, the set of maps S represents the relations,
GGA
(a) gives a necessary condition for the induced adjunction Re : U C/S DG
G ⊥ G M : Sing to be
a Quillen adjunction.
Recall from Theorem 2.3.1 that the Bousfield (left) localization U C/S shares the same
id
underlying category with U C, the localization map U C GGA U C/S is the identity, and the
8
See the section 1.2 or the beginning of the section 3.4 for more explanations.
99
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
GGA
identity functors id : U C DG
G ⊥ G U C/S : id form a Quillen pair. There is therefore an induced
GGA
adjunction Re : U C/S DG
G ⊥ G M : Sing, where Re is already known for sending cofibrations
to cofibrations because the localization U C/S has the same cofibrations as U C. The big
picture is the diagram
C
r
id
UC
⊥
U C/S
id
Sing
Re
a
Re
a
Sing
γ
M.
Not every model category admits a small presentation, but there is a large class of
categories that do : the combinatorial model categories (definition 2.1.6), introduced by Jeff
Smith.
Theorem 3.2.3. Any combinatorial model category admits a small presentation.
Proof. This theorem is proved in [Dug01a].
It is proven in Theorem 6.1 of [Dug01b] that any left proper combinatorial model category
can be replaced with a Quillen equivalent simplicial model category. Since a universal model
category U C is about as nice as we could expect, proper, combinatorial, simplicial, . . . , we
can get rid of the properness assumption, and even having it as result.
Corollary 3.2.4. Any combinatorial model category is Quillen equivalent to a left proper
simplicial model category.
Proof. By theorem 3.2.3, there is a small presentation U C/S of the given combinatorial
model category. Since U C is in particular left proper and simplicial and these properties
are inherited by any left Bousfield localization, we get that U C/S is also left proper and
simplicial.
3.3
Local Model Structures on Simplicial Presheaves
op
The homotopy cocompletion of a small category r : C ,GGA sSetC does not take account
of the colimits present in the indexing category C, in the sense that the embedding r does
not commute with colimits, it just formally adds them9 . Endowing the resulting category
op
sSetC with a global model structure, either the two extremal structures, the injective or
the projective, or any intermediate such as the flasque model structure, defines a homotopy
op
theory on sSetC , exploiting the homotopy theory of sSet.
9
See Section 2.3 in [Dug] for some more explanations.
100
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
op
To keep track of the colimits in C that should appear in sSetC , a solution is to endow the
category C with a Grothendieck topology10 : a collection of families {Uα GGA X}α required
to satisfy reasonable axioms to model the families of open covers from topology. Indeed, the
colimits we want to keep will be the ones that appear in these covering families. However,
there are several different ways to transport the data encoded in the Grothendieck topology
op
to the category of simplicial presheaves sSetC , all of them leading to Quillen equivalent
model structures. They are usually called local model structures, and all share the same local
op
weak equivalences of sSetC and thus, they all are models for the same homotopy category.
We will now give the usual definition of local weak equivalences of simplicial presheaves,
and will after switch to the characterization due to Dugger and Isaksen.
Simplicial sets have homotopy groups, which are the homotopy groups of the realized
topological space. That is, given a simplicial set X• with a choice of base point x0 ∈ X0 , if
we denote its realization by |X• |, its homotopy groups are defined by
πn (X• , x0 ) := πn (|X• |, x0 ),
which are abelian group objects in Set for n ≥ 2 and only group objects for n = 1. A map
f
X• GGA Y• of simplicial sets is a weak equivalence if and only if
• the induced function π0 (X• ) GGA π0 (Y• ) is a bijection (of sets);
• the induced homomorphisms πn (X• , x0 ) GGA πn (Y• , f (x0 )) are isomorphisms of groups
for n ≥ 1 and all choices of base points x0 ∈ X0 .
There is an alternate version in which there is no choice of base point, by taking the disjoint
union on all possible base points. Define for any n ≥ 1 and any simplicial set X• ∈ sSet
the object
a
πn (X• ) :=
πn (X, x0 ),
x0 ∈X0
which is a group objects in Set, with a function
πn (X• ) GGA X0
induced by all the functions πn (X• , x0 ) GGA X0 . This definition is functorial and a map of
simplicial sets X• GGA Y• ∈ sSet induces a commutative square for every n ≥ 1
πn (X• )
πn (Y• )
X0
Y0 .
f
Moreover, such a morphism X• GGA Y• ∈ sSet is a weak equivalences of simplicial sets if
and only if
10
Grothendieck topologies are defined in section 1.3.
101
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
• it induces a bijection π0 (X• ) GGA π0 (Y• ) of sets;
• the induced squares
πn (X• )
πn (Y• )
X0
Y0
y
are pullback squares for every n ≥ 1.
Since all the construction is functorial, it extends to simplicial presheaves, i.e., diagrams of
op
simplicial sets. Given a simplicial presheaf F ∈ sSetC , define
π0 (F ) : C op GGA Set : C G
[ GA π0 (F (C)),
and similarly for any n ≥ 1 define
πn (F ) : C op GGA Set : C G
[ GA πn (F (C)),
op
and functoriality gives morphisms πn (F ) GGA F0 ∈ SetC . However, these homotopy
groups for simplicial presheaves do not take account of the topology of the category C. To
give a local meaning to πn (F ), we will use the sheafification functor
(−)++ : Pre(C) GGA Sh(C).
op
Definition (Local Model Structures on sSetC ). A morphism of simplicial presheaves
f
F GGA G ∈ sSetC
op
is called a local weak equivalence if
∼
=
• the induced morphism π0 (F )++ GGA π0 (G)++ is an isomorphism of sheaves;
• the induced squares
πn F ++
πn G++
F0++
G++
0 ,
y
are pullback squares for every n ≥ 1.
Denote by L the class of local weak equivalences of simplicial presheaves, and fix a global
op
model structure on sSetC , either the projective of the injective. The local model structure
associated is the (left) Bousfield localization, when it exists, at the class of local weak
equivalences L.
102
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
op
The word local is not chosen by chance. Indeed, if the topos sSetC has enough points,
a morphism of simplicial presheaves F GGA G is a local weak equivalence if and only if
it induces weak homotopy equivalences on all stalks Fx GGA Gx , see for example Remark
6.5.2.2 [Lur09] for more explanation.
Since the class L is in general not a set but a proper class, we cannot directly apply the
machinery of Bousfield localization, but rather need to construct it. We will now mention
three different constructions of such a local homotopy category, and then focus with more
details on a particular one.
Historically, it is Joyal who first constructed a model category structure whose homotopy
category gives the local model structure, in [Joy84]. Unlike the other local model structures
we will give, Joyal did not produce a model structure by refining a global model structure
op
on the category of simplicial presheaves sSetC , but rather by focusing on the subcategory
of simplicial sheaves, denoted by sSh(C). This model structure, denoted by sSh(C)Joyal or
sSh(C)inj can now be called the local injective model structure on sSh(C). Indeed, cofibrations
are objectwise cofibrations, weak equivalences are the local weak equivalences, and the
fibrations are defined by the right lifting property with respect to cofibrations that are local
weak equivalences. The advantage of only working with sheaves is that they already respect
the covers coming from the covering sieves, since a sheaf F sends coequalizers
a
GGA
Uα ×X Uβ GGA
a
Uα → X,
GGA
F (Uα ) GGA
Y
F (Uα ×X Uβ ).
to equalizer diagrams
F (X) →
Y
The second method was introduced by Jardine, in the form of a local injective model
op
structure on sSetC . He further proved, in Theorem 5.9 [Jar11], that it gives the same
homotopy category as Joyal’s model. Indeed, the sheafification functor and the embedding
of sheaves in presheaves form a Quillen equivalence for these two models
C op
GGA
(−)++ : sSh(C)Joyal DG
G ⊥ G sSetJardine : i.
These models are very close : Jardine’s model also has objectwise cofibrations as cofibrations,
local weak equivalences as weak equivalences and fibrations defined by the lifting property.
Therefore, this model can be seen as the left Bousfield localization of the global injective
op
model on sSetC at this class L of local weak equivalences
op
op
sSetCJardine = sSetCinj /L.
The local projective version, as the Bousfield localization with respect to L, also exists.
The proof, based however on the existence of the injective local model, can be found as
Theorem 1.6 in [Bla01]. Furthermore, Isaksen proved similar statements in [Isa05] using his
interemediate model structure, the flasque model structure.
103
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
3.3.1
Hypercompletion
More recently, a completely different approach to get to these local model structures was
discovered by Dugger, Hollander and Isaksen. This method is called the hypercompletion,
summarized in Section 7 of [Dug01c], and explained with proofs in [DHI04]. We will give
an explicit construction and show that it is Quillen equivalent to the local model structures
obtained by localization with respect to L. One advantage of this method is that it is given
as a small presentation, as defined in the last section. Moreover, this method may be seen
as more ’universal’ than the others. Indeed, the construction is inspired by a property that
topological spaces enjoy. Roughly speaking, an open hypercover of a topological space X is
a simplicial space11 U∗ such that
• each Un is a coproduct of representables;
• U0 is an open cover of X;
• U1 is an open cover of double intersections of U0 ;
• U2 is an open cover of triple intersections of U1 ;
and so on. An important property of topological spaces is that their homotopy type can be
recovered as a homotopy colimit of any of its hypercover.
Theorem 3.3.1. Let X be a topological space. If U∗ is an open hypercover of X, then the
natural map
hocolim U∗ GGA X
is a weak equivalence.
Proof. More details and a complete proof can be found in [DI04a].
The procedure involved in this section is called the hypercompletion, which is the localization with respect to the maps hocolim U∗ GGA rX, where U∗ GGA rX runs through
op
the hypercovers of sSetC . Since the injective local model structure of Jardine has recently served as the foundation from which Morel and Voevodsky built their A1 -homotopy
theory for schemes in [MV99], we will state all the results also starting from the injecop
tive model structure on sSetC . However, all the results also hold in the projective setting
op
U C = sSetCproj or in an intermediate global model structure. As usual, since this localization
is in general not with respect to a set, but rather by a proper class, the general machinery
does not apply, and we need an explicit construction.
Before starting on the construction, let’s summarize all the model structures already
presented, with the authors and the references where they appeared or were proved first.
Moreover, the projective, injective and flasque versions are (at least) left proper, cellular
and simplicial. Indeed, their global structure is left proper, cellular and simplicial, and by
Theorem 4.1.1 [Hir03], these properties still hold after applying a left Bousfield localization12 .
11
12
A simplicial space is a simplicial object in Top, as defined in section 1.4.
In fact, these are also right proper, see the paragraph after Definition 4.1 in [Isa05].
104
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
intermediate
injective
global
Heller [Hel88]
projective
flasque
others
Isaksen [Isa05]
?
Bousfield-Kan [BK72]
Hirschhorn-Quillen [Hir03]
local :
Joyal (S) [Joy84]
localization wrt L
Jardine [Jar87]
Isaksen [Isa05]
Jardine [Jar06]
Brown-Gersten (S) [BG73]
Blander [Bla01]
local :
Dugger-Hollander-Isaksen [DHI04]
hypercompletion
Table 3.1: The (S) means that the model structure is on the subcategory of simplicial
sheaves sSh(C).
Recall that the global model structures are defined for any indexing small category C, while
the local structures require a Grothendieck topology on the small category C. However, the
global model structures may also be seen as local model structures where the small category
C is endowed with the chaotic topology, i.e., the topology generated by the pretopology
containing only isomorphisms as covering families. In this case, the simplicial sheaves are
also the same as the simplicial presheaves.
Back to the construction of the hypercompletion localization, again inspired by the
situation of topological spaces, we will use a characterization of ’local morphisms’, with the
language of ’local’ lifting properties. This was developed by Dugger and Isaksen, and is
explained in details in [DI04b].
Let’s fix a Grothendieck site C with a choice of (pre)topology. Given a morphism of
simplicial presheaves F GGA G, a map of simplicial sets K GGA L and an object of the site
op
X ∈ C, a commutative square in sSetC
K ⊗ rX
F
L ⊗ rX
G
is said to have local liftings, if there is a covering sieve R ,GGA X such that for any ’open
subset’ of the sieve Uα ,GGA X, the refinement
K ⊗ rUα
K ⊗ rX
F
L ⊗ rUα
L ⊗ rX
G
admits a lifting. A morphism F GGA G ∈ sSetC
diagrams
105
op
is called a local fibration if all the
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Λk [n] ⊗ rX
F
∆[n] ⊗ rX
G
op
admit local liftings for all objects X ∈ C. A simplicial presheaf F ∈ sSetC is called locally
op
fibrant if the unique map F GGA ∗ is a local fibration. Morevover, call F GGA G ∈ sSetC
a local acyclic fibration or a local trivial fibration if it is both a local fibration and a local
weak equivalence. Their characterization in terms of local liftings is given in the following
proposition.
Proposition 3.3.2 (Characterization of local acyclic fibrations). A map of simplicial presheaves
f
F GGA G ∈ sSetC
op
is a local acyclic fibration if and only if the diagrams
∂∆[n] ⊗ rX
F
f
∆[n] ⊗ rX
G
have local liftings for all objects X ∈ C.
Proof. This is Proposition 7.2 in [DI04b].
An immediate consequence is that local acyclic fibrations are closed under pullbacks.
Indeed if
F ×G H
F
H
G
y
op
is a pullback square in sSetC , the lifting in the diagram
∂∆[n] ⊗ rU
∂∆[n] ⊗ rX
F ×G H
F
∆[n] ⊗ rU
∆[n] ⊗ rX
H
G,
induces a desired lifting ∆[n] ⊗ rU GGA F ×G H by the universal property of the pullback.
Another useful corollary that we will use, is the following.
106
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
op
Corollary 3.3.3. Let F GGA G ∈ sSetC be a local fibration (resp. local acyclic fibration)
and let K ,GGA L be a monomorphism of finite simplicial sets (inclusion at each level).
Then, the induced map
op
F L GGA F K ×GK GL ∈ sSetC
is a local fibration (resp. local acyclic fibration).
The local liftings from proposition 3.3.2 are in general not compatible between the different maps Uα ,GGA X of the sieve, and so the Uα ’s cannot be assembled into a simplicial
op
object in sSetC that covers X and that admits a unique lifting ∆[n]⊗Uα GGA F . However,
one can arrange for this kind of compatibility by using hypercovers.
Definition (Hypercover, refinement). Let X ∈ C be an object of the site. A hypercover of
op
X is a simplicial presheaf U with a map U GGA rX ∈ sSetC such that
op
(1) each Un = α rUn,α ∈ SetC is a coproduct of representables;
(2) the map U GGA rX is a local acyclic fibration.
`
A refinement of a hypercover U GGA rX, is another hypercover V GGA rX that factors
op
through U GGA rX in sSetC , as in the diagram
V
rX.
∃
U
Observe that hypercovers can be seen as augmented simplicial objects in presheaves
augmented by constant presheaves. The first condition says that a hypercover
models the role of covers by open subsets, as when the site in question is the small site
top(X) of open subsets of X. To better understand the second condition, we need to
[C op , sSet],
f
op
introduce the notion of a generalized cover. A map E GGA B ∈ SetC between presheaves
is called a generalized cover (sometimes a local epimorphism) if for all maps rX GGA B,
there is a covering sieve R ,GGA X such that any map Uα GGA X lifts through f as in the
diagram
E
∃
rUα
rX
f
B.
The word ’generalized’ is here to remind us that this is not really a cover in the category C,
`
but only a cover at the level of presheaves. However, whenever B ∈ C and E = α rEα is
`
a coproduct of representables, a map α rEα GGA rB is a generalized cover if and only if
the sieve generated by {Eα GGA B}α is a covering sieve of B.
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
op
The simplicial enrichment of sSetC defines an object13 F ∂∆[n] ∈ sSetC
cial presheaf C G
[ GA F (C)∂∆[n] that sends
op
as the simpli-
m G
[ GA sSet(∂∆[n] × ∆[m], F (C)) ∈ Set.
fn F the C-diagram which is the 0th object of F ∂∆[n] , seen as a simplicial object
Denote by M
op
in SetC . This is the presheaf defined by
fn F : C op GGA Set : C G
M
[ GA sSet(∂∆[n], F (C)).
fn F induced
∼ sSet(∆[n], F ) GGA sSet(∂∆[n], F ) = M
Note that there is a natural map Fn =
∆[n]
∂∆[n]
by F
GGA F
, in its turn induced by the simplicial inclusion ∂∆[n] ,GGA ∆[n].
Proposition 3.3.2 may now be rephrased to give the following alternate characterization of
local acyclic fibrations.
Lemma 3.3.4 (Characterization of local acyclic fibrations). A map of simplicial presheaves
f
F GGA G ∈ sSetC
op
is a local acyclic fibration if and only if the natural maps
C op
fn F ×
Fn GGA M
en G Gn ∈ Set
M
are generalized covers for all n ≥ 0.
f
Proof. By proposition 3.3.2, a map F GGA G is a local acyclic fibration if and only if for
every object of the site X ∈ C, any diagram
∂∆[n] ⊗ rX
F
f
∆[n] ⊗ rX
G
admits a local lifting, i.e., there is a covering sieve R ,GGA X such that the diagram
∂∆[n] ⊗ rUα
∂∆[n] ⊗ rX
F
∃θ
∆[n] ⊗ rUα
f
∆[n] ⊗ rX
θ
admits a lifting ∆[n] ⊗ rUα GGA F . Using the fact that sSetC
G
op
is tensored and cotensored
θ
over sSet and using the Yoneda lemma, a map ∆[n] ⊗ rUα GGA F is the same data as
θ̃
a map rUα GGA Fn , and similarly for rUα GGA Gn . Again by adjointness, the map
fn F , and similarly for G. The existence
∂∆[n]⊗rUα GGA F is the same as a map rUα GGA M
θ
of a simplicial filler ∆[n] ⊗ rUα GGA F is similar to the existence of fillers rUα GGA Fn for
op
all n ≥ 0 in the following diagrams sitting in SetC
13
See the section 3.4 for more explanations.
108
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Fn
∃θ̃
fn F
M
rUα
fn
rX
fn F ×
M
e n G Gn
M
fn G.
M
Gn
Indeed, the fact that the diagram
Fn
fn
θ̃
rUα
Gn
rX
commutes, represents the fact that
F
θ
∆[n] ⊗ rUα
f
∆[n] ⊗ rX
commutes, and similarly for
Fn
θ̃
rUα
fn F
M
rX
and
109
G
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
∂∆[n] ⊗ rUα
∂∆[n] ⊗ rX
F.
θ
∆[n] ⊗ rUα
By the pushout property, it is therefore equivalent to have the required liftings
Fn
θ
rUα
fn F ×
M
e
M
rX
nG
Gn ,
fn F ×
that is, the maps Fn GGA M
en G Gn are generalized covers.
M
op
A map U GGA rX ∈ sSetC is therefore a local acyclic fibration if and only if U0 GGA
fn U are generalized covers for n ≥ 1.
X, U1 GGA U0 ×X U0 , and more generally Un GGA M
In particular, if {Uα GGA X}α is a covering family, the associated C̆ech complex
···
`
α0 ,α1 ,α2
Uα0 ×X Uα1 ×X Uα2
`
α0 ,α1
Uα0 ×X Uα1
`
α0
U α0
X
is a hypercover, where we set the n-th term to be the coproduct of all fibered products
a
Uα0 ×X · · · ×X Uαn ,
α0 ,...,αn
and didn’t draw the degeneracies for convenience. Such a C̆ech complex is a nicely behaved
∼
=
fn U are not only generalized covers but isomorhypercover because all the maps Un GGA M
phisms. In a general hypercover, the n-th term Un contains all the fibered products from
Un−1 , but may be further refined itself, by taking a generalized cover. The local liftings in
proposition 3.3.2 were not required to be compatible for the different maps Uα ,GGA X of
a sieve. However, up to taking a hypercover, these local liftings can now be arranged to be
compatible.
f
op
Proposition 3.3.5. Let F GGA G ∈ sSetC be a local acyclic fibration, K GGA L ∈ sSet
be a monomorphism of finite simplicial sets, and let X be an element of the site C with a
hypercover U GGA rX. Given any commutative square
K ⊗U
F
L⊗U
G,
there exists a refinement V GGA X of the hypercover U GGA X such that the extended
diagram
110
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
K ⊗V
K ⊗U
F
L⊗U
G,
∃
L⊗V
admits a lifting.
Proof. By using the general natural isomorphisms of the tensored and cotensored sSetop
enriched structure on sSetC
op
op
sSetC (K ⊗ U, F ) ∼
= sSetC (U, F K ),
the three morphisms
K ⊗ U GGA F,
K ⊗ U GGA G,
and
L ⊗ U GGA G
are the same as three morphisms
U GGA F K ,
U GGA GK
and
U GGA GL .
Moreover, the data of the commutative square is equivalent to a morphism U GGA F K ×GK
GL . We want a refinement of the hypercover
V
rX.
∃
U
such that the following diagram admits a lifting
FL
V
F K ×GK GL .
U
By Corollary 3.3.3, the vertical map is a local acyclic fibration. The idea is to construct
more generally such refinements for any local acyclic fibration, which is done in the following
lemma 3.3.6 and proposition 3.3.7.
Such a refinement is constructed inductively on the n-th skeleton of U∗ . The core of the
induction step is the following lemma.
op
op
Lemma 3.3.6. Let F GGA G ∈ SetC be a generalized cover, and Y GGA G ∈ SetC be
any morphism of presheaves of sets. Then there exists a generalized cover V GGA Y such
that V is a coproduct of representables and such that there is a lifting
111
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
F
V
Y
G.
Proof. The generalized cover V GGA Y is constructed by taking the coproduct on ’all
f
op
possible candidates’. For all object X ∈ C and any morphism rX GGA Y ∈ SetC , since
rX GGA Y GGA G is a map from a representable, there exists a covering sieve Rf such
that for any map U GGA X in the sieve there is a lifting in the diagram
F
rU
rX
Y
G.
Let now V be the coproduct on all possible such maps

a
V =

a
U .

U →X∈Rf
f
rX →U
This forces the induced map V GGA Y to be a generalized cover, and since in the coproduct
every element admits a lifting, the diagram
F
V
Y
G
also admits the desired lifting.
As defined after Definition 1.4.1 for a complete and cocomplete category M, the preop induces an n-truncation functor on simplicial
composition by the inclusion ∆op
n ,GGA ∆
objects
M∆
op
trn
op
GGA M∆n ,
that has a left adjoint by left Kan extension
∆op
fn : M∆n GGA
sk
: trn ,
DG
G ⊥G M
op
called the n-skeleton, as well as a right adjoint given by right Kan extension
trn : M∆
op
∆n
g n,
GGA
: cosk
DG
G ⊥G M
op
called the n-coskeleton. By abuse of language, the composites
fn ◦ trn : M∆
skn = sk
op
GGA M∆
op
and
112
g n ◦ trn : M∆
coskn = cosk
op
op
GGA M∆ ,
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
op
are also called the n-skeleton and the n-coskeleton. Given a simplicial object W ∈ M∆ ,
define the (n + 1)st latching object and the (n + 1)st matching objects by
Ln+1 W := (skn W )n+1
Mn+1 W := (coskn W )n+1 .
and
fn a trn and the unit of trn a cosk
g n give two maps of simplicial
The counit of the adjunction sk
objects
skn W GGA W GGA coskn W,
which give the natural maps
Ln+1 W GGA Wn+1 GGA Mn+1 W
in the (n + 1)st level.
Before getting back to the proof of our inductive step, let’s define some terminology.
Using the language of augmented simplicial objects, as given in Definition 1.4.2, turns out
to be useful while working with hypercovers; it allows hypercovers U GGA rX to be seen as
objects themselves rather than as morphisms. Given an object M ∈ M, denote an action
of a set S ∈ Set on it by
Y
M ·S :=
M.
s∈S
For an augmented simplicial object W : ∆+ GGA M and an augmented simplicial set
·K
K : ∆+ GGA Set, the end of the functor W(−)(−) is given by the equalizer

hom+ (K, W ) := eq 

Y
GGA
Wn·Kn GGA
n≥−1
Y
·Kk 
Wm
∈ M.
k→m
For an augmented simplicial object W ∈ M∆+ , there are useful isomorphisms, see for
example Section 4 in [DHI04]
(coskn W )k ∼
= hom+ (∆k , coskn W ) ∼
= hom+ (skn ∆k , W ),
which shows in particular when k = n + 1 that the (n + 1)-st matching object of W is
Mn+1 W = (coskn W )n+1 ∼
= hom+ (∂∆[n], W ).
The inductive step of the construction of the refinement we seek, is finally given in the
op
following lemma. A morphism V GGA rX ∈ sSetC is an n-truncated hypercover if the
op
induced map skn ◦ trn V GGA skn ◦ trn rX ∈ sSetC is a hypercover. The following technical
lemma is taken from Proposition 5.5 in [DHI04].
op
Proposition 3.3.7. Let F GGA G ∈ sSetC be a local acyclic fibration, for X ∈ C let
op
U GGA rX be a hypercover and let U GGA G ∈ sSetC be a morphism of simplicial
presheaves. Fix an integer n ≥ 0 and suppose there exists an n-truncated hypercover V GGA
rX refining skn U and that admits a lifting
113
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
F
V
U
G.
Then there exists an (n+1)st-truncated hypercover W GGA rX refining skn+1 U that admits
a lifting
F
W
U
G,
and such that the n-skeleton of this whole diagram agrees with the n-skeleton of the preceding
diagram.
Proof. By taking the pullback
F0
y
F
loc.acyc.fib.
V
U
n-hyp.
G
loc.acyc.fib.
rX,
we will need to produce an (n + 1)-hypercover W GGA rX that admits a lifting W GGA F 0 .
Since local acyclic fibrations are closed under pullback, see after Proposition 3.3.2, the
vertical composite F 0 GGA rX is a local acyclic fibration. By Corollary 3.3.3, the induced
morphism
F 0∆[n+1] GGA F 0∂∆[n+1] ×rX ∂∆[n+1] rX ∆[n+1]
is also a local acyclic fibration. If n > 0 (the case n = 0 being slightly different) since
fn+1 F , and since
∼M
∂∆[n + 1] and ∆[n + 1] are connected we have the isomorphism Mn+1 F =
in addition rX is a discrete simplicial presheaf, we have the isomorphisms rX ∂∆[n+1] ∼
= rX ∼
=
∆[n+1]
0∆[n+1]
0∂∆[n+1]
rX
. So the preceding induced map is a local acyclic fibration F
GGA F
,
which by lemma 3.3.4, gives in level 0 a generalized cover
0
Fn+1
GGA Mn+1 F 0 .
So there is an induced diagram
114
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
0
Fn+1
gen.cover
Mn+1 F 0 ,
Mn+1 V
op
and by lemma 3.3.6, there exists a coproduct of representables presheaves Z ∈ SetC , with
a generalized cover Z GGA Mn+1 V that admits a lifting
0
Fn+1
∃
Z
Mn+1 V
Mn+1 F 0 .
Ln+1 V
We can consider the hypercover W GGA rX given by trn W := trn V in which we add the
`
(n + 1)-st term by Wn+1 = Z Ln+1 V . Recall that Ln+1 V is just the (n + 1)-st term of its
skeleton (skn V )n+1 .
This induction construction finishes the proof of proposition 3.3.5. This will allow the
op
identification we are interested in : the hypercompletion of sSetC (localization with respect
to hocolim U GGA rX for all hypercovers U GGA rX) is the same as the localization of
op
sSetC with respect to local weak equivalences. However, since hypercovers do in general
not form a set, the localization is confronted with set-theoretic issues. A way to avoid this
size issue is to prove the following stronger result.
op
Call a collection S of hypercovers in sSetC dense, if every hypercover U GGA rX in
op
sSetC may be refined by a hypercover V GGA rX which belongs to S. We will state the
op
general result it in its injective version because of the foundational importance of sSetCJardine ,
but it also holds in the flasque or the projective setting.
op
Theorem 3.3.8. Let S be a class of hypercovers in sSetCinj that contains a set which is
op
dense. The left-Bousfield localization sSetCinj /S, called hypercompletion, exists and coincides
with Jardine’s injective local model structure sPre(C)Jardine .
Proof. Let L be the collection of all local weak equivalences, and recall that Jardine’s model
op
is the left-Bousfield localization sSetCinj /L = sPre(C)Jardine . In order to compare the two
localizations, we will use the criterion developed in lemma 2.3.2. Since S is possibly not a
op
set, the localization of sSetC at S may not exist. However, if S is a set, the localization
op
exists by theorem 2.3.1 since sSetC is left proper and cellular. Two cases may occur.
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
case 1 : Suppose first that S is itself a set. Since by definition S ⊆ L, if we prove that any
a local acyclic S-fibration between S-fibrant objects is indeed an S-equivalence, we are
done by lemma 2.3.2. The verification of the hypothesis of lemma 2.3.2 is proved in the
following lemma 3.3.9.
case 2 : Let now S 0 be a dense set of hypercovers contained in S. So case 1 says that
op
op
sSetC /S 0 = sSetC /L, i.e., every local weak equivalence is an S 0 -weak equivalence.
In particular, since S ⊆ L, every hypercover in S is an S 0 -weak equivalence. Since the
op
op
converse is also true, this shows that sSetC /S exist and is the same as sSetC /S 0 .
In order to finish the proof, we need to check the hypothesis of lemma 2.3.2, this is done
in the following lemma 3.3.9. Again, all this work for any choice of model structure on
op
op
sSetC , but we keep sSetCinj for convenience. If S is a set of hypercovers, an S-fibration
op
op
in sSetCinj is a fibration in the left Bousfield localization sSetCinj /S, and similarly for an
S-fibrant object.
f
op
Lemma 3.3.9. Let S be a set of hypercovers in sSetCinj that is dense, and let F GGA G ∈
op
sSetCinj be an S-fibration and a local weak equivalence, between S-fibrant objects. Given any
object of the site X ∈ C, every square
∂∆[n] ⊗ rX
F
∃
∆[n] ⊗ rX
G
admits a lifting. In particular, the map f is an objectwise acyclic fibration and therefore an
S-equivalence.
f
Proof. Since S ⊆ L, the S-fibration F GGA G is also an L-fibration, i.e., a local fibration.
But it also is a local weak equivalence by hypothesis, so by using that rX GGA rX is a hypercover (it is an isomorphism), Proposition 3.3.2 says that there exists another hypercover
U GGA rX that admits a lifting
∂∆[n] ⊗ U
∂∆[n] ⊗ rX
F
∃
∆[n] ⊗ U
∆[n] ⊗ rX
G.
Moreover, since S is dense, there exists a refinement V GGA rX ∈ S, and we could work in
the diagram
116
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
∂∆[n] ⊗ V
∂∆[n] ⊗ U
∂∆[n] ⊗ rX
F
∆[n] ⊗ V
∆[n] ⊗ U
∆[n] ⊗ rX
G,
where the lifting is just the composite of the initial lifting with ∆[n] ⊗ V GGA ∆[n] ⊗ U .
So without loss of generality, let’s assume for convenience that the hypercover U GGA rX
is already in S.
op
Remark that all the different model structures on sSetC are simplicial model structures,
the property being inherited by working with diagrams of simplicial sets, and let as before
op
Map(−, −) : n G
[ GA sSetC (− × ∆[n], −)
op
denote the natural homotopy function complex of the simplicial model structure sSetCinj /S.
Since F GGA G is an S-fibration between S-fibrant objects, and since ∂∆[n] ,GGA ∆[n]
is a cofibration of simplicial sets, the induced morphism
F ∆[n] GGA F ∂∆[n] ×G∂∆[n] G∆[n]
is an S-fibration, which also is a local weak equivalence by corollary 3.3.3. Moreover, note
that U GGA rX is an S-equivalence, U and rX are cofibrant (everything is cofibrant in the
injective model), and both F ∆[n] and F ∂∆[n] ×G∂∆[n] G∆[n] are S-fibrant. All this induces
the following diagram of simplicial mapping spaces
∼S
Map(X, F ∆[n] )
Map(U, F ∆[n] )
S-fib
S-fib
Map(X, F ∂∆[n] ×G∂∆[n] G∆[n] )
∼S
Map(U, F ∂∆[n] ×G∂∆[n] G∆[n] ),
where the horizontal maps are S-equivalences and the vertical arrows are S-fibrations. Our
initial commutative square
∂∆[n] ⊗ rX
F
∆[n] ⊗ rX
G
represents a 0-simplex in the lower left corner x ∈ Map(X, F ∂∆[n] ×G∂∆[n] G∆[n] ). The desired
lifting ∆[n] ⊗ X GGA F is given by a lift in the upper left corner. The existence of a lifting
∆[n] ⊗ U GGA F in a previous diagram, may be rephrased as the fact that the image of x
in the lower right corner has a lift in the upper right corner. Since the two horizontal maps
are weak equivalences, there is at least a 0-simplex y in the connected component of x that
117
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
admits a lifting in the upper left corner. But fibrations of simplicial sets are surjective onto
the connected components of their image, so x also admits a lift in the upper left corner as
desired.
op
Corollary 3.3.10. The left-Bousfield localization of sSetCinj with respect to all hypercovers
U∗ GGA rX gives the same model category as Jardine’s injective local model structure
op
sPre(C)Jardine = sSetCloc.inj .
Proof. The collection of all hypercovers contains a dense set, see Proposition 6.7 in [DHI04].
The proof relies on the fact that the Grothendieck site is small. The set containing all
hypercovers of ’size’ less than the cardinality of the set of morphisms of C, is a indeed a
dense set.
3.3.2
Characterization of cofibrant and fibrant objects
We refer to the Section 7 in [DHI04] for some applications of the construction of the local
model structure as a localization with respect to hypercovers. Even though the local injective
model structure of Jardine has been used for the foundation of A1 -homotopy theory, the
projective version, with it’s universality as described in the previous section 3.2, turns out to
be often more useful. Moreover, the projective model admits a much nicer characterization
of the fibrant objects. In order to characterize them, we need the following definition.
op
Definition (Descent condition for presheaves). Let sSetC be endowed with a global model
op
structure. An objectwise fibrant simplicial presheaf F ∈ sSetC satisfies descent for a
op
hypercover U GGA rX ∈ sSetC , if the natural map from F (X) to the homotopy limit of
the diagram
a
a F (U0 )
Q
a
a F (U1 )
a
a F (U2 )
Q
Q
···
is a weak equivalence, where the products range over the representable summands of U . A
op
op
simplicial presheaf F ∈ sSetC satisfies descent for a hypercover U GGA rX ∈ sSetC if
some objectwise fibrant replacement of F does.
∼
Note that this definition has been arranged so that if F GGA G is an objectwise weak
equivalence of simplicial presheaves, F satisfies descent for some hypercover U GGA rX if
and only if G does. Moreover, this condition is very similar to the condition that a presheaf
has to satisfy to actually be a sheaf. Indeed, given a pretopology on the Grothendieck site
C, a presheaf F is a sheaf if and only if for any covering family {Ui GGA X}i in the site,
the natural map


F (X) GGA eq 
Y
F (Ui ) ⇒
i
Y
F (Ui ×X Uj )
i,j
is an isomorphism, i.e., F (X) is the coequalizer. So a preasheaf F satisfies descent for all
hypercovers if and only if it is a sheaf up to homotopy. There is a characterization of the
fibrant objects, due to Dugger, Hollander and Isaksen.
118
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Corollary 3.3.11 (Characterization of fibrant objects in local model structures). Let S be
a collection of hypercovers that contains a dense set.
op
• A simplifial presheaf F ∈ sSetCinj /S is fibrant in the Bousfield localization if and only if
op
it is fibrant in the global injective model sSetCinj and satisfies descent for all hypercovers
in S;
op
• A simplifial presheaf F ∈ sSetCproj /S is fibrant in the Bousfield localization if and only
if it is objectwise fibrant and satisfies descent for all hypercovers in S.
In particular, F satisfies descent for S if it satisfies descent for all hypercovers.
Proof. It is Corollary 6.3 in [DHI04].
3.4
The Category of (Nisnevich) Motivic Spaces
Throughout all the section, let’s fix a noetherian scheme S of finite Krull dimension. These
conditions are not really necessary for our purpose, but some further characterizations require them14 . Denote by Sm/S the category of smooth schemes of finite type over S, which
is a full subcategory of the category Sch/S of all schemes over S, and given by
• Objects : U → S smooth morphism of finite type;
f˜
f
• Morphisms : (U → S) GGA (V → S) is a morphism of schemes U GGA V such that
the triangle commutes
U
f˜
S.
V
The objects in this category will often be identified with the underlying scheme U they
f˜
represent, and the morphisms with the morphisms between the underlying schemes U GGA
V . It has an initial object, the empty scheme ∅ → S, as well as a terminal object, the
identity morphism S → S. The goal of this section is to set a homotopy theory for these
schemes, which will be done by using the machinery introduced in the previous section.
A reason why we restrict our attention to schemes of finite type is that this condition
assures us that the category Sm/S is essentially small. Moreover, an important goal of this
motivic homotopy is ultimately to be able to study more concrete objects such as varieties
over fields, rather than arbitrary schemes. However, the restriction to smooth schemes of
Morel and Voevodsky’s work in [MV99] may be more for convenience. Indeed, important
results in A1 -homotopy theory such as the Purity Theorem15 3.2.23 in [MV99], only holds
for smooth schemes.
For example, these properties ensures that the topos MS with the local model structure coming from
the Nisnevich topology is locally of finite homotopy dimension, see Section 7.2 in [Lur09] for more on this.
15
The analogue of the tubular neighbourhood theorem for A1 -homotopy theory.
14
119
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
3.4.1
The category of motivic spaces
In order to set a good place in which to have an interesting homotopy theory of schemes,
we will first homotopy cocomplete the category Sm/S, as it is done in section 3.1. In fact,
the categories of schemes Sm/S are do not contain all the (homotopy) colimits and this is
a way to remedy to this problem. This is given by embedding it in the functor category
(Sm/S)op ,GGA [(Sm/S)op , Set] ,GGA [(Sm/S)op , sSet],
first by the Yoneda embedding and then by taking the discrete simplicial set on it. This
category is called the category of motivic spaces on S and denoted by MS .
Before starting doing homotopy theory in this category, let’s study its categorical structure. Most of it is what is objectwise inherited by the structure already present in sSet.
These properties are essentially the ones listed at the beginning of chapter 2 in [DRØ03].
First, note that it is cocomplete, by construction. It turns out it also is complete, since
sSet is and limits can be computed objectwise in functor categories. Moreover, it is locally
finitely presentable by being a free cocompletion, as shows Theorem 1.46 in [AR94]16 . This
means that there is a set A (which are here the representables), such that each object of
the category can be obtained as a directed colimit of objects of A, and such that for each
object A ∈ A, the hom-functor
A(rA, −) : A GGA Set
preserves direct colimits. That is, every object is built up on this pieces, see Proposition
3.2.1 and 3.2.2. As a main consequence, finite limits commute with (small) directed colimits,
as shows Proposition 1.59 in [AR94].
Consider now the category sSet of simplicial sets as a closed strict symmetric monoidal
category with the categorical product as the monoidal product, the constant simplicial set
{∗}∗ as the unit, and the internal-hom given by the bifunctor
sSet(−, −) : sSetop × sSet GGA sSet,
where we define
sSet(K, L) : n G
[ GA sSet(K × ∆[n], L).
The tensor-hom adjunction that turns it into a closed monoidal category are the natural
isomorphisms
sSet(K × L, M ) ∼
= sSet(K, sSet(L, M )).
This monoidal structure induces again a closed strict symmetric monoidal structure on MS ,
by objectwise product
F, G ∈ MS
=⇒
(F × G)(U ) := F (U ) × G(U ) ∈ sSet,
and the unit is the constant (discrete) simplicial presheaf I(U ) = {∗}∗ ∈ sSet. Moreover it
is a closed monoidal structure with the internal-hom bifunctor
MS (−, −) : MSop × MS GGA MS ,
16
This uses the fact that the category of smooth schemes Sm/S is essentially small.
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
that is again defined objectwise by means of the internal-hom of sSet
MS (F, G) : U G
[ GA sSet(F (U ), G(U )) ∈ sSet.
The adjunction relation follows from the preceeding one, and so there are natural isomorphisms
∼ MS (F, MS (G, H)).
MS (F × G, H) =
By viewing a simplicial set K ∈ sSet as a (constant on objects) motivic space
K : (Sm/S)op GGA sSet : U G
[ GA K,
there also is a product bifunctor (the tensor), that is denoted by
− ⊗ − : MS × sSet GGA MS : (F, K) G
[ GA F ⊗ K,
and defined by (F ⊗ K)(U ) = F (U ) × K ∈ sSet. This product gives the sSet-enrichment
of MS by means of the bifunctor
MapMS (−, −) : MSop × MS GGA sSet,
which is defined by
MapMS (F, G) : n G
[ GA MS (F ⊗ ∆[n], G).
This will give in addition a homotopy function complex for MS , when it will be endowed
with the structure of a model category. There is also a functorial cotensoring with the
bifunctor
(−)(−) : MS × sSetop GGA MS ,
defined by (F K )(U ) = F (U )K = sSet(K, F (U )) ∈ sSet. Note that both this tensoring and
cotensoring operations give the same result if the simplicial set is embedded in MS as a
constant (on objects) simplicial presheaf, and the operation is performed in MS .
To summarize, MS is a bicomplete, locally finitely presentable and closed strict symmetric monoidal category, with the objectwise structure from sSet and the adjunction
MS (F × G, H) ∼
= MS (F, MS (G, H)).
It can also be seen as enriched over sSet which for F ∈ MS and K ∈ sSet gives the
tensoring F ⊗ K ∈ MS and cotensoring F K ∈ MS that satisfy the adjunction relations
MapMS (F ⊗ K, G) ∼
= sSet(K, MapMS (F, G)) ∼
= MapMS (F, GK ).
3.4.2
Homotopy theory on motivic spaces
The category of motivic spaces MS can be first endowed with any of the global model
structures of section 3.1. These are the global injective and the global projective model
structure, both with the same weak equivalences and therefore giving the same homotopy
category Ho(MS ).
The global injective model structure MS,inj is given by
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
• cofibrations : the monomorphisms of simplicial presheaves, i.e., object-wise monomorphisms of simplicial sets;
• weak equivalences : the object-wise weak equivalences of simplicial sets;
• fibrations : the morphisms of simplicial presheaves that have the right lifting property
with respect to all acyclic cofibrations.
Every simplicial presheaf is cofibrant in this model, and there is no good characterization
of fibrant objects. Moreover, this is a proper, cellular, simplicial model category where the
generating sets of (acyclic) cofibrations are hard to describe. They are subsets of all (acyclic)
cofibrations, where the whole class is restricted to a generating set by a size argument.
The global projective model structure MS,proj is given by
• fibrations : the object-wise (Kan) fibrations of simplicial sets;
• weak equivalences : the object-wise weak equivalences of simplicial sets;
• fibrations : the morphisms of simplicial presheaves that have the left lifting property
with respect to all acyclic cofibrations.
In this model, the representables rX are still cofibrant (Proposition 3.1.12) and fibrant
objects are the object-wise Kan complexes. In particular, representables are also fibrant in
this model. Moreover, this also is a proper, cellular, simplicial model category where the
generating set of cofibrations is given by
I = {∂∆[n] ⊗ X ,GGA ∆[n] ⊗ X}
n∈N
X∈Sm/S
,
and the generating set of acyclic cofibrations is given by
n
J = Λk [n] ⊗ X ,GGA ∆[n] ⊗ X
o
k≤n,n>0
X∈Sm/S
.
From Proposition 3.2.2, this model admits the universal property that any functor Sm/S GGA
M into a model category M factorizes (uniquely) through MS,proj
Sm/S
r = yoneda
MS,proj
=⇒
Re a Sing
γ
M,
∼
where, the natural transformation Re ◦ r =⇒ γ is a natural weak equivalence and the
GGA
adjunction Re : MS,proj DG
G ⊥ G M : Sing is a Quillen adjunction.
In particular, in both model, cofibrations are object-wise monomorphisms and fibrations
are object-wise Kan fibrations. By definition of these models, the identity adjunction
GGA
id : MS,proj DG
G ⊥ G MS,inj : id
122
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
is a Quillen adjunction. Unfortunately, these two models are not very interesting model
structures for at least two reasons. First, they do not take account of the colimits already
present in Sm/S, since the embedding Sm/S ,GGA MS of homotopy cocompletion does not
preserve homotopy colimits. Second, some colimits are ’geometrically wrong’ in the initial
category of schemes and should be corrected in the motivic category. Such an example,
coming from Example 2.1.1 in [Dug], is the following. Consider the diagram of affine schemes
over a field k
A1 − {0}
A1
A1
where the embeddings are z G
[ GA z and z G
[ GA 1/z. By the usual contravariant equivalence
between algebra and geometry, the pushout of schemes corresponds to the pullback of kalgebras
k[z]
k[z −1 ]
k[z, z −1 ],
which is just the ground field k. The above pushout of affine lines is therefore the point
scheme Spec k. However, by considering the affine line to be something like a real line,
geometric intuition would prefer this pushout to be a projective line. The problem may be
formulated as that the colimit of the underlying topological spaces is not the underlying
topological space of the colimit.
This issue is fixed by localizing the global model structure on MS to recover exactly the
’geometric colimits’ we want to have in our category. The resulting model structure is a
local model structure and there are different ways to achieve it, as explained in the previous
chapter in section 3.3. More precisely, we will adopt here the method by hypercompletion,
because it’s the method that explains the best the main idea of recovering our colimits. The
colimits to be recovered will be the ones encoded in the Grothendieck topology that Sm/S
will be endowed with.
3.4.3
The category of Nisnevich motivic spaces
There are many choices for topologies on schemes, see for example [Sta, Chapter : Topologies
on Schemes] for a bunch of them. The two most well-known are the Zariski topology and the
étale topology. Roughly speaking, the Zariski topology is the one in which the open covers
are given by the scheme-theoretic open immersions (which are also jointly surjective). This
is the analogue of the small site top(X) of open subsets of X, for a topological space X.
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Unfortunately, this topology is not well-suited for A1 -homotopy theory and gives unexpected
and unwanted results, in particular because it does not contain enough covers (asking for
an open immersion is too much, there are not enough open subsets). A coarser topology
with more open covers is the étale topology, where the open covers are defined by the étale
morphisms (that are jointly surjective. The étale topology has already proved its utility
throughout the étale cohomology of schemes. However, this topology has too many open
covers and again can give some unexpected results. For example, a field which is non
separably algebraically closed may have a non-trivial étale cohomology, see for example
Remark 17.9 in [Mil80].
We will now define the completely decomposed version of the étale topology, the Nisnevich
topology, which is in some sense what we get by forcing fields to be acyclic. The Nisnevich
topology, which is strictly finer that the Zariski topology and strictly coarser that the étale
topology, seem to enjoy many good properties of both topologies, while avoiding some of
their defects. We refer to the introduction of chapter 3 in [MV99] for more profound reasons
about this choice of topology in A1 -homotopy theory.
In what follows, we will denote a scheme (X, OX ) by X, the stalk at a point x ∈ X by
OX,x and its residue field by k(x).
f
Definition (Completely Decomposed Morphism). A morphism of schemes U GGA X is
said to be completely decomposed at x ∈ X if there exists u ∈ U such that f (u) = x and
such that the residual field extension k(x) GGA k(u) is an isomorphism.
Note that such a point u ∈ U corresponds to a unique filler in the diagram
U
f
Spec k(x)
∗ 7→ x
X.
Moreover, in the situation of a pullback square
V
f˜
Y
y
g
U
f
X,
f˜
f
if U GGA X is completely decomposed at some x = g(y), then its pullback V GGA Y is also
completely decomposed at y. Indeed, a filler Spec k(x) GGA U in
124
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
V
U
y
∴∃
Spec k(y)
∃
Spec k(x)
f˜
f
Y
X
g
induces the required filler Spec k(y) GGA V by the universal property of the pullback. A
covering in the Nisnevich topology will be a (surjective) family of completely decomposed
étale morphisms.
fi
Definition (Nisnevich Coverings). A finite set of morphisms {Ui GGA S} in Sm/S is called
a Nisnevich covering if each fi is étale (of finite type) and if for each point x ∈ S there
fi0
exists an index i0 such that Ui0 GGA S is completely decomposed at x.
Since a Zariski covering {Ui ,GGA S} only contains open embeddings Ui ,GGA S and any
point in Ui has the same residual field as its image in S, Zariski covers are Nisnevich covers.
Moreover, any Nisnevich cover is in particular an étale cover (that is in addition completely
decomposed).
To show that this generates a Grothendieck topology on Sm/S, observe first that clearly
∼
=
an isomorphism of schemes {S 0 GGA S} is Nisnevich covering. Moreover, since the property
of being an étale morphism is closed under pullbacks along any other morphism and the
pullback of a completely decomposed morphism is again a completely decomposed morphism
by the argument above, the Nisnevich covers are closed by change of basis. Finally, suppose
gij
fi
that {Ui GGA S}i is a Nisnevich covering, and that {Vij GGA Ui }j are Nisnevich coverings
for all i. We need to see that {Vij GGA Ui GGA S}i,j gives a Nisnevich covering. First,
the composition of étale morphisms of finite type is again an étale morphism of finite type.
Also, given any point x ∈ S, there is a u ∈ Ui0 such that fi0 (u) = x and such that
∼
=
k(x) GGA k(u) is an isomorphism. Similarly, there is a v ∈ Vi0 j0 such that gi0 j0 (v) = u
∼
=
and again k(u) GGA k(v) is an isomorphism. Therefore, this same v ∈ Vi0 j0 satisfies
∼
=
∼
=
fi0 ◦ gi0 j0 (v) = x and k(x) GGA k(u) GGA k(v), therefore {Vij GGA Ui GGA S}i,j is a
Nisnevich covering.
We may now invoke the machinery of section 3.3 to get local model structures on MS .
Both the projective and the injective global structures may be used, so we can always use
whichever suits best the situation. Recall that the local model is the left Bousfield localization of the global model, with respect either to local weak equivalences, or only to the
hypercovers from the site (which are acyclic fibrations). The cofibrations in the local model
will be the same as in the global model, and the weak equivalences are the local weak equivalences17 . The fibrations are identified, by Corollary 3.3.11, with the simplicial presheaves
Which are the ones inducing isomorphisms on all sheaves of homotopy groups, or, since the topos MS
has enough points, the ones inducing isomorphisms on all stalks.
17
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
F that are fibrant in the global model, and which admit descent for all hypercovers (or any
class of hypercovers which contains a dense set).
More precisely, the local injective model MS,loc.inj has
• cofibrations : the monomorphisms of simplicial presheaves;
• weak equivalences : the local weak equivalences;
• fibrations : the morphisms having the right lifting property with respect to (local)
acyclic cofibrations.
This model still has the nice property that every object is cofibrant, while fibrant objects
are the objects that were fibrant in the global injective model and that satisfy descent for
all hypercovers (Corollary 3.3.11), i.e., the fibrant objects of the global injective model that
are (Nisnevich) sheaves up to homotopy.
The local projective model MS,loc.proj has
• cofibrations : the same as in MS,proj ;
• weak equivalences : the local weak equivalences;
• fibrations : the morphisms having the right lifting property with respect to (local)
acyclic cofibrations.
Since the cofibrations are the same, the representables rX are still cofibrant, while fibrant
objects are the objects that were fibrant in the global projective model and that satisfy
descent for all hypercovers (Corollary 3.3.11). Thas is, the fibrant objects are the objectwise Kan complexes that are (Nisnevich) sheaves up to homotopy. Since for any smooth
scheme X ∈ Sm/S, its embedding rX ∈ MS,loc.proj is a sheaf in the étale topology, then
it also is in the coarser Nisnevich topology. Therefore the local projective model has the
advantage that, for any scheme X, its image rX ∈ MS,loc.proj is both cofibrant and fibrant.
Moreover, both local models are left proper, simplicial and cellular since the left Bousfield
localization preserves these properties (Theorem 2.3.1). In fact, they are also right proper,
see for example Lemma 1.7 in [Bla01] for the projective version.
3.5
Unstable Motivic Homotopy Theory
We now have two local model structures on the category of motivic spaces MS , where
the local models reflect some aspects of the ’geometry’ of the initial category Sm/S of
smooth schemes of finite type over S. Indeed, the localization process from the global model
structures to the local model structures may be seen as a way of correcting how colimits are
computed in MS to give them a geometric meaning, as illustrated in the previous section 3.4.
However, is this localization enough to give an interesting homotopy theory for schemes ?
Thanks to the property that any representable rX is cofibrant and fibrant in the local
projective model structure on MS , the answer is that the category of motivic spaces does
not help us identify schemes.
Proposition 3.5.1. The Yoneda embedding from the category of smooth schemes Sm/S to
the homotopy category of motivic spaces Ho(MS ), is fully faithful.
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
Proof. The idea is to use the local projective model, and the functor that embeds schemes
in the homotopy category of simplical presheaves is the composite
r
Sm/S ,GGA MS,loc.proj GGA Ho(MS,loc.proj ).
Since in this model, for any schemes X, Y ∈ Sm/S, their images rX, rY ∈ MS,loc.proj is both
cofibrant and fibrant, the hom-set in the homotopy category is the quotient of the hom-set
in the model category by the homotopy relation
Ho(MS )(rX, rY ) = MS,loc.proj (rX, rY )/ ∼ .
Moreover, since ∆[0] ,GGA ∆[1] is an acyclic cofibration of simplicial sets, then rX ⊗ ∆[0] ∼
=
rX GA rX ⊗ ∆[1] is a cofibration of simplicial presheaves. Therefore, by the 2-out-of-3
property the unique morphism
id ⊗!
∼ rX ∼
rX ⊗ ∆[1] GGGA rX ⊗ ∗ =
= rX ⊗ ∆[0]
is also a weak equivalence. Moreover, the cofibration ∂∆[1] ,GGA ∆[1] induces a cofibration
rX
a
∼ rX ⊗ ∂∆[1] GA rX ⊗ ∆[1],
rX =
and thus rX ⊗ ∆[1] is a cylinder object for rX
rX
`
rX
rX
∼
rX ⊗ ∆[1].
By Corollary 1.2.6 in [Hov99], since rX is cofibrant and rY is fibrant, for any homotopic
GGA rY , there is a homotopy through any cylinder object, so in particmaps f ∼ g : rX GGA
ular through rX ⊗ ∆[1]. However, since rX and rY are discrete, there are only constant
homotopies and thus f ∼ g =⇒ f = g. This shows that Sm/S embeds fully faithfully in
Ho(MS ) by the chain of isomorphisms
Sm/S(X, Y ) ∼
= MS (rX, rY ) ∼
= MS,loc.proj (rX, rY )/ ∼ = Ho(MS,loc.proj )(rX, rY ).
The local model structure on motivic spaces MS are therefore no enough for studying
the category of schemes Sm/S. To try to adjust these model structures, we can apply the
same construction to Grothendieck sites that are more geometric than Sm/S, and see what
goes out of it. There are two important examples, which give respectively a homotopy
theory of topological spaces and a homotopy theory of real manifolds, and which lead both
to the same conclusion : there still is a missing ingredient.
These examples are the Example 5.6 in [Dug01c] for the topological spaces, and the
example of real manifolds is given in Section 8 of the same article. Given a small category
C, recall that U C is the homotopy theory built from C, as defined in section 3.2. Observe
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
that starting with C = ∗, tautologically U C gives sSet, the homotopy theory of simplicial
sets. However, simplicial sets are usually seen as ’built out of ∆’, so a more interesting
example would be by starting with the category of simplices ∆. Proposition 3.2.2 gives a
Quillen adjunction in the diagram
r
∆
n
7→
∆n
U∆
a
Top,
which commutes only up to a natural transformation. It turns out that the adjoint pair
GGA
U ∆ DG
G ⊥ G Top is not exactly a Quillen equivalence, the only missing ingredient is that there
is no information in U ∆ which says that the representables ∆[n] are contractible. Indeed,
if we denote the set A = {∆[n] GGA ∗}, the Quillen pair descends to a Quillen equivalence
of model categories
GGA
U ∆/A DG
G ⊥ G Top.
Another interesting example which gives a similar conclusion is by starting with the category
GGA
ManR of real manifolds. Similarly, there is a Quillen pair U ManR DG
G ⊥ G Top, which
becomes a Quillen equivalence of model categories, after localizing with respect to the maps
π
A = {R × M GGA M }
GGA
U ManR /A DG
G ⊥ G Top.
The moral of the story is that the this universal construction needs to know something
about the interval ∆[1], or the real line R being contractible. A similar process is applied in
the category of motivic spaces, where an interesting homotopy theory arises where the role
of the interval or the real line is played by the affine line A1 .
In some sense one could interpret the situation as the following. Endowing the category
of motivic spaces with a local model structure is just a way to set up a good category that
reflects the geometry of schemes. Identifying A1 as an interval and localizing to make it
behave like an interval is a way to set up a place in which one can in addition do homotopy
theory of schemes.
In the two previous examples, the fact that the interval was contractible automatically
∼
forces I × F GGA F into weak equivalences. A direct approach in the motivic setting would
be to apply a (left) Bousfield localization with respect to all those morphisms rA1 × F GGA
F . However, this is in general not a proper class and therefore existential problem for this
localization may occur.
A first step would be to declare the morphism rA1 GGA ∗, which can be identified with
the morphism rA1 × ∗ GGA ∗, to be a weak equivalence, and then see which maps are forced
to become weak equivalences in the localized category. Recall that what is denoted here
id
by ∗ is the representable simplicial presheaf associated to the object S GGA S ∈ Sm/S,
the terminal object of the category of schemes. When we consider Sm/k, the category of
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
smooth schemes of finite type over a field k, this terminal object is the identity between the
scheme with one point Spec k = {∗}. Observe first that any morphism ∗ GGA A1 becomes a
weak equivalence in the localized category by the two out of three property of the diagram
∼
rA1
∗
∗.
id
As a consequence, any morphism of schemes ∗ GGA A1 ∈ Sm/S will turn into a weak
∼
equivalence ∗ GGA rA1 in the category of motivic spaces MS . Note that when the initial
category of schemes is Sm/k, such a map Spec k GGA A1 corresponds exactly to a k-rational
point of the affine line A1 . Moreover, for any fibrant simplicial presheaf F ∈ MS , one also
get that the canonical projection
∼
rA1 × F GGA F
has to be a weak equivalence. Indeed, since the local models are right proper, the pullback
rA1 × F
F
y
rA1
fib.
∼
∗
∼
implies that the projection A1 × F GGA F has to be a weak equivalence too.
More generally, if rA1 is fibrant in some local model structure on MS , and if F is a
fibrant replacement functor in this model, for any simplicial presheaf G ∈ MS , consider the
following glueing of pullback squares
rA1 × G
G
y
∼
rA1 × F(G)
F(G)
y
fib.
rA1
∼
fib.
∼
∗.
The morphism rA1 × F(G) GGA F(G) is a weak equivalence by right properness, and also
∼
a fibration since rA1 GGA ∗ is. Therefore, the vertical morphism rA1 × G GGA rA1 × F(G)
is also a weak equivalence, again by right properness, and the upper square becomes
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CHAPTER 3. MOTIVIC HOMOTOPY THEORY
rA1 × G
∼
G
y
∼
∼
rA1 × F(G)
F(G).
∼
By the two out of three property, the canonical projection rA1 ×G GGA G is then also forced
to be a weak equivalence. We can now use the local projective model structure MS,loc.proj
in which the representable rA1 is fibrant. This implies that all the canonical projections
rA1 × F GGA F would become weak equivalences if we localize the local projective model
structure with respect to a map ∗ GGA rA1 .
Definition (Motivic unstable model category). Given a local model structure on MS , the
left Bousfield localization at the morphism rA1 GGA ∗ gives an unstable motivic model
1
1
category, denoted here by MSA , or just by M A when there is no confusion about the base
scheme.
Since the injective and the projective local model structures share the same weak equivalences, the canonical projections rA1 × F GGA F are weak equivalences in any of the
unstable motivic category. By formally applying a left Bousfield localization, we proved the
following theorem.
Theorem 3.5.2. The left Bousfield localization of the local projective or local injective model
structure on the category of motivic spaces MS at the canonical projections rA1 × F GGA F
1
gives the unstable motivic model structure MSA .
Therefore, the cofibrations in the unstable motivic model are the same as in the local
models, which are the same as in the global model. In particular, the representable functors
rX, for any scheme X ∈ Sm/S, are cofibrant in both models, and any motivic space
1
F ∈ MSA is cofibrant in the injective model. The weak equivalences in this model are
called A1 -weak equivalences, and are given by the Bousfield localization. Recall that for two
motivic spaces F, G ∈ MS , there is a notion of simplicial mapping space given by
Map(F, G) : n G
[ GA MS (F ⊗ ∆[n], G).
Call S = {rA1 ×F GGA F } the class of natural projections, where we can use either notation
rA1 × F = A1 ⊗ F , and let’s work in the local projective model MS,loc.proj . A fibrant motivic
space G ∈ MS,loc.proj is called A1 -local (or S-local) if for any such projection, the induced
map
∼
Map(F, G) GGGA Map(F × rA1 , G)
∈ sSet
is a weak equivalence of simplicial sets. The A1 -local objects are exactly the fibrant objects
A1
in the Bousfield localization MS,loc.proj
. Moreover, a morphism of motivic space G GGA H
is called an A1 -weak equivalence, if for every A1 -local object F , the induced map
∼
Map(H, F ) GGGA Map(G, F )
130
∈ sSet
CHAPTER 3. MOTIVIC HOMOTOPY THEORY
is a weak equivalence of simplicial sets. By the general machinery, the A1 -weak equivalences
are exactly the weak equivalences in the Bousfield localization.
131
Index
λ-sequence, 50
I-cell complex, 52
C̆ech complex, 110
acyclic cofibration, 37
acyclic fibration, 37
associativity coherence, 12
associator, 11
augmentation map, 32
augmented simplex category, 32
augmented simplicial object, 32, 113
bifibrant, 37
closed symmetric monoidal category, 13
cocylinder object, 43
cofibrant, 37
cofibrant replacement, 44
cofibrantly generated model category, 54
cofibration, 36
combinatorial model category, 62, 63
compact object, 59
completely decomposed, 124
copower, 19
cosimplicial object, 30
cosimplicial resolution, 95
coskeleton, 31, 112
cotensor, 18
cotensored category, 18
covering family, 21
covering sieve, 21
cylinder object, 42
derived functor, 46
descent condition, 118
effective monomorphism, 60
enriched category, 11, 16
fibrant, 37
fibrant replacement, 45
fibration, 36
flasque model structure, 89
generalized cover, 107
geometric realization, 10
global injective model structure, 81
global projective model structure, 81
Grothendieck pretopology, 21
Grothendieck topology, 21
homotopy category, 42
homotopy function complex, 71
hypercompletion, 104
hypercover, 107
indiscrete topology, 26
internal-hom, 13–15
left Bousfield localization, 73
left derived functor, 46
left homotopy, 43
left Kan extension, 9
left lifting property, 36
left localization, 72
left proper, 63
left Quillen functor, 47
local acyclic fibration, 106
local epimorphism, 107
local equivalence, 73
local fibration, 105
local injective model structure, 103
local lifting, 105
132
INDEX
local model structure, 101
local object, 73
local weak equivalence, 102
locally fibrant, 106
locally finitely presentable, 120
locally finitely presentable category, 62
locally presentable category, 62
mapping space, 65
model category, 36
monoidal category, 11
motivic space, 120
Nisnevich covering, 125
Nisnevich topology, 124
power, 19
presheaf, 24, 29
proper, 63
small object argument, 49
small presentation, 99
small site, 21
symmetric monoidal category, 12
symmetry coherence, 12
tensor, 18
tensor product, 11
tensored category, 18
total left derived functor, 47
total right derived functor, 47
transfinite composition, 50
truncation functor, 31, 112
unit coherence, 12
unstable motivic model category, 130
weak equivalence, 36
weak factorization system, 39
Quillen adjunction, 47
Quillen equivalence, 48
Quillen pair, 47
recognition theorem, 55
relative I-cell complex, 52
retract, 34
retraction, 35
right derived functor, 46
right homotopy, 44
right Kan extension, 9
right lifting property, 36
right localization, 72
right proper, 63
right Quillen functor, 47
sheaf, 25
sheafification functor, 27
simplicial category, 65
simplicial object, 30
simplicial presheaf, 19, 67
singular functor, 10
site, 21
skeleton, 31, 112
small object, 50
133
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