Download An Introduction to Model Categories and Abstract Homotopy Theory

An Introduction to Model Categories and
Abstract Homotopy Theory
Frank Gounelas
February 10, 2009
Contents
1 Some Notes on Category Theory
2
2 Classical Homotopy Theory
4
3 Model Categories
5
4 Abstract Homotopy Theory
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1
Introduction
The aim is to introduce model categories and generalise the notions of homotopy theory that
one has in the classical topological setting. The first section aims to recall some facts needed
from category theory, my references for this were the book by Kashiwara and Schapira [KS06]
but also the notes I took from a course of Rouquier [Rou08]. The second section recalls the
setting of homotopy in topology, defining the homotopy extension and lifting properties and
briefly discussing homotopy equivalences in this settings, mentioning also what a fibration and
cofibration is in the category of topological spaces. My reference for this was Baues’s book
[Bau89]. The third and fourth sections introduces model categories and abstract homotopy
theory as per Quillen, reinterpreted and expanded by Hovey [Hov99] and Dwyer and Spalinski
[DS95].
1
Some Notes on Category Theory
Definition 1.1. Let I be a small category and let F : I → C be a functor. A cone of F is
an (N, φ) where N ∈ Ob(C) and for every X ∈ I, φX : N → F (X) such that for every arrow
f : X → Y in I we have that the following diagram commutes
yy
yy
y
y
y| y
F (X)
φX
N E
EE
EEφY
EE
E"
/ F (Y )
F (f )
Definition 1.2. Let I be a small category and let F : I → C be a functor. A cocone of F
is an (N, φ) where N ∈ Ob(C) and for every X ∈ I, φX : F (X) → N such that for every arrow
f : X → Y in I we have that the following diagram commutes
/ F (Y )
F (X)
EE
F (f )
z
EE
z
EE
zz
z
EE
z φ
φX
|zz Y
"
N
Definition 1.3. A limit (colimit) is a terminal (initial) object in the category of cones (cocones).
One can interpret pushouts and pullbacks using limits and colimits arising from the small categories I1 = {A ← B → C} and I2 = {A → B ← C} respectively which have three objects and
two non-identity morphisms. Namely, one has
Definition 1.4. A functor F : I1 → C is given by a diagram F (A) ← F (B) → F (C) in C.
The colimit of F is called the pushout P of this diagram. We have a commutative square, such
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that for each f1 , f2 as below, there exists a unique f making the entire diagram commute.
F (B)
i
/ F (C)
j
j′
F (A)
f1
/P
′
i
f
f2
," Y
Definition 1.5. A functor F : I2 → C is given by a diagram F (A) → F (B) ← F (C) in C.
The limit of F is called the pullback P of this diagram. We have a commutative square, such
that for each f1 , f2 as below, there exists a unique f making the entire diagram commute.
F (B) o
O
i
F (C)
O ]
j′
j
F (A) o
f
i′
P b
f2
f1
f
Y
A small category is one such the collection of objects of this category forms a set. One says
that a category C has all (small) limits or colimits if for every (small) category I and functor
F : I → C, the limit or colimit of F exists.
By considering the trivial functor ∅ → C and taking the limit or colimit, one can trivially conclude
the following.
Lemma 1.6. If a category C has all limits (or colimits), then C has a terminal (or initial
respectively) object.
One might want to invert (make into isomorphisms) a particular family of morphisms in a
category C in a process which resembles the localisation of a ring. To do this, one starts with
a set S of morphisms and forms the big category C[S −1 ] with various universal properties as
follows.
Definition 1.7. Let C be a category and S a family of morphisms of C. A localisation of C
at S, is a big category C[S −1 ] and a functor F : C → C[S −1 ] such that
1. For all f ∈ S, F (f ) is an isomorphism.
2. For any other big category and functor G : C → A such that G(s) is an isomorphism for
all s ∈ S, there exists a functor GS : C[S −1 ] → A and an isomorphism G ∼
= GS ◦ F .
3. If G1 and G2 are two functors C[S −1 ] → A then there is a bijection
HomFunct(C[S −1 ],A) (G1 , G2 ) ↔ HomFunct(C,A) (G1 ◦ F, G2 ◦ F )
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The universal properties guarantee as usual that the big category C[S −1 ] is unique up to equivalence of categories.
One might also want to consider identifying two families of morphisms as the same thing, in a
process generalising taking a quotient by an equivalence relation for a group for example.
Definition 1.8. For a category C, let ∼ denote an equivalence relation which is defined
to consist of for every objects X, Y in C, an equivalence relation ∼ on HomC (X, Y ) such that
if f ∼ f ′ for f, f ′ : X → Y , if g : Y → Z and g ′ : Z ′ → Xfor some objects Z, Z ′ in C, then
gf ∼ gf ′ and f g ′ ∼ f ′ g ′ . Given such a relation, one can form the quotient category C/ ∼
which has the same objects as C, but HomC/∼ (X, Y ) = HomC (X, Y )/ ∼.
Note that there is a canonical quotient functor C → C/ ∼.
2
Classical Homotopy Theory
Throughout this section we let I = [0, 1] denote the unit interval.
Definition 2.1. Two maps f, g : X → Y between topological space are homotopic f ∼ g if
there is a map H : X × I → Y such that Hi0 = f and Hi1 = g where ij : X → X × I where
i0 : X → X × {0} and i1 : X → X × {1}.
Composition between homotopies between maps remains a homotopy and so given more so that
homotopy is an equivalence relation one can form the homotopy category Ho = Top / ∼ by
the categorical quotient of Top by this equivalence relation. One say that a map f : X → Y
between topological spaces is a homotopy equivalence if there exists a g : Y → X such that
f g ∼ 1Y and gf = 1X .
Definition 2.2. We say that a map i : A → X has the homotopy extension property
with respect to another topological space Y if for every diagram below, there exists an H such
that everything commutes
< X JJJ
f
yy
JJJi0
y
JJ
yy
y
JJ
y
$
yy
_H_ _/8& Y
X
A EE
: ×I
u
EE
u
EE
uu
E
uui×idI
u
i0 EE
g
"
uu
A×I
i
One says that the map i : A → X is a cofibration if it satisfies the homotopy extension property
with respect to any other topological space Y .
Definition 2.3. We say that a map p : X → B has the homotopy lifting property with
respect to another topological space Y if for each diagram as below, there exists an H making
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the diagram commute
f
Y
/X
x<
x
xx
p
i0
xxH
x
x
/B
Y ×I
G
One says that p : X → B is a fibration if it has the homotopy lifting property with respect to
any other topological space Y .
A map f : X → Y between topological spaces is called a weak homotopy equivalence if it
induces an isomorphism πn (X, x) → πn (Y, f (x)), where x is a basepoint of X, for n ≥ 1 and
simply a bijection for n = 0.
Definition 2.4. A map p : X → Y is called a Serre fibration if p has the homotopy lifting
property with respect to every CW-complex A.
3
Model Categories
Model categories have three distinguished classes of maps, namely
∼
• Weak equivalences denoted as −
→
• Fibrations denoted as ։
• Cofibrations denoted as ֒→
each of which is closed under composition and also contains all identities.
Definition 3.1. A model category is a category C with three classes of morphisms as above,
such that
1. C has all small limits and colimits
2. (2 out of 3) If f, g are maps in C such that gf is defined, then if two out of f, g, gf are
weak equivalences, so is the third
3. (Retracts) If f is a retract of g and g is fibration/cofibration/weak equivalence, then so is
f . Diagrammatically, a retract is given as follows
X
i
/Y
f
g
X′
/ Y′
i
such that ri = idX and r′ i′ = idX ′ .
5
r
/X
f
r
/ X′
∼
4. (Lifting) Trivial cofibrations (namely cofibrations which are also weak equivalences ֒−
→)
have the left lifting property with respect to fibrations and cofibrations have the left lifting
property with respect to trivial fibrations. Diagrammatically, for every commutative square
as below, there exists an h such that
f
A _
i ∼
~
B
h
~
~
g
f
A _
/X
~>
p
i
/Y
~
B
h
~
~
g
/X
~>
p ∼
/Y
5. (Factorisations) Any map f : A → C can be factored in the following two ways f : A ֒→
∼
∼
B −−−։ C and f : A ֒−
→B։C
From the first axiom and lemma (1.6), each model category has initial and terminal objects and
so it makes sense to also define a fibrant (cofibrant) object as that for which the map to the
terminal object (from the initial object) is a fibration (cofibration respectively).
Example 3.2. The category Top of topological spaces has model structures on it. For example
one can take f : X → Y to be a
• weak equivalence if it is a weak homotopy equivalences
• fibration if it is a Serre fibration
• cofibration if it is a retracts of a map X → Y ′ where Y ′ is obtained from X by attaching
cells.
Example 3.3. For a ring R, the category ChR of non-negatively graded chain complexes over
R and a morphism f : M → N is a
• weak equivalence if it induces an isomorphism on the homology groups
• fibration if for all k ≥ 1, fk : Mk → Nk is surjective
• cofibration if for all k ≥ 0, fk : Mk → Nk is injective, with a projective R-module are the
cokernel.
Example 3.4. If C is a model category, then so is C op . For an object A in a model category
C, the category of objects under (A ↓ C) and the category of objects over A (C ↓ A) are also
model categories.
4
Abstract Homotopy Theory
Definition 4.1. Let C be a model category and let f : A → X a morphism in C.
1. A cylinder object
` for A is an object A ∧ I of C such that there exists a factorisation of
the fold map A A → A into a cofibration and a weak equivalence as follows
a
∼
A
A ֒→ A ∧ I −
→A
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inj
Note that we have two maps i0 , i1 : A → A ∧ I by composing A −−→ A
i is the cofibration.
`
i
A−
→ A ∧ I, where
2. A path object for X is an object X I of C such that there exists a factorisation of the
diagonal map X → X × X into a weak equivalence and a fibration as follows
∼
X−
→ XI ։ X × X
p
prj
Note that we have two maps p0 , p1 : X I → X by composing X I −
→ X × X −−→ X where p
is the fibration.
3. Two maps f, g`: A → X are left homotopic if there exists a cylinder object A ∧ I such
that f + g : A A → X factors as
a i0 +i1
H
A −−−→ A ∧ I −→ X
A
where H is called the homotopy and Hi0 = f and Hi1 = g.
4. Two maps f, g : A → X are right homotopic if there exists a path object X I such that
(f, g) : A → X × X factors as
H
(p0 ,p1 )
A −→ X I −−−−→ X × X
where H is called the homotopy and p0 H = f and p1 H = g.
5. Two maps f, g : A → X are homotopic f ∼ g if they are both left and right homotopic.
6. A map f is a homotopy equivalence if there exists a map h : X → A such that f h ∼ 1X
and hf ∼ 1A .
The notation A ∧ I and X I aims to generalise that of definition (2.1). In fact, Quillen uses A × I,
but this can lead to confusion with the product operation in the category C, since in our case, I
is not an object of C, but more of a symbol to distinguish between the various possible cylinder
or path objects.
Definition 4.2. For a general category C, define Ho(C) as C localised at the weak equivalences,
that is C[W −1 ] if W is the class of weak equivalences.
Note that Ho(C) is not necessarily a category (but a big category, see (1.7)) in the conventional
sense, since it is possible that HomHo(C) (A, B) does not form a set. It is however the case
that when C is a model category, the morphisms do form a set and Ho(C) is in fact called the
homotopy category (This is an application of theorem (4.3)).
Localisations of categories are hard to describe in general. What makes the model structure one
can impose on a category so important is the following theorem.
Theorem 4.3. Let C be a model category and Ccf the full subcategory of C whose objects are
both cofibrant and fibrant objects in C. There is an equivalence of categories
Ho C
∼
/ Ho Ccf ∼
= Ccf / ∼
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References
[Bau89] Hans Joachim Baues. Algebraic homotopy, volume 15 of Cambridge Studies in Advanced
Mathematics. Cambridge University Press, Cambridge, 1989.
[DS95] W. G. Dwyer and J. Spaliński. Homotopy theories and model categories. In Handbook
of algebraic topology, pages 73–126. North-Holland, Amsterdam, 1995.
[Hov99] Mark Hovey. Model categories, volume 63 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 1999.
[KS06] Masaki Kashiwara and Pierre Schapira. Categories and sheaves, volume 332 of
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2006.
[Rou08] Raphael Rouquier. Applied Homological Algebra. Notes from a graduate course at
Oxford University. 2008.
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