Download RATIONAL HOMOTOPY THEORY Contents 1. Introduction 1 2

RATIONAL HOMOTOPY THEORY
JONATHAN CAMPBELL
Contents
1. Introduction
2. Building Spaces
2.1. Postnikov Towers
3. As Much as I Can Say About Spectral Sequences
3.1. Two Degree Filtration
3.2. Three Degree Filtration
3.3. Exact Couples
3.4. General Filtrations
3.5. Misc.
4. The Serre Spectral Sequence
5. Some Rational Computations
6. Serre Theory
7. Rational Homotopy Groups of Spheres
8. Rational Spaces and Localization
8.1. Rational H-spaces
9. DGAs
9.1. Basic Definitions
9.2. Sullivan Models
9.3. Minimal Sullivan Models
10. Appendix: Simplicial Sets
10.1. Geometric Realization and Model Structure
11. Appendix: Model Categories
11.1. The Small Object Argument
11.2. Cofibrantly Generated Model Categories
References
1
2
2
5
6
7
8
9
9
10
11
12
14
15
17
18
18
19
21
23
25
27
31
34
35
1. Introduction
Resources: Griffiths-Morgan, Felix-Halperin-Thomas.
Should probably say something about prequisites. One should know
• Algebraic topology as in Hatcher (counting the homotopy theory chapter).
• Basic spectral sequences
There are also things it would be useful to know, but which I’ll review and/or create handouts on
• simplicial sets / simplicial objects in a category C
• model categories
We’ll start off with Serre theory, which is a computational tool. While it is a computational tool, it gives
us some indication that operating only with certain filters on our eyes can immensely simplify things. In
particular, it hints that if we were to consider only the parts of spaces that rational (co)homology sees, life
is a little bit easier.
1
This is where the philosophy of homotopy theory actually comes in. We think of homotopy theory
as trying to classify all shapes up to the relation of “homotopy” which is our usual notion of being able
to deform without tearing, etc. However, “modern” homotopy theory is really a rich theory of how to
consider difference objects equivalent. For example, we can consider two objects equivalent if they are
homotopy equivalent, or two chain complexes equivalent if they are quasi-isomorphic. There are even
looser notions of equivalence, for example, two spaces are Q-equivalent if their rational homologies are
equivalent.
Using the tools of homotopy theory, we can examine what kind of theory we get if we allow ourselves
such a notion of equivalence. In fact, what we can show is that if we restrict ourselves to only caring about
rational equivalence, the category of topological spaces is completely algebraic. That is, it is equivalent to
some category completely determined by algebraic data (some elaboration of a category of algebras).
The goal in this course is to approach the problem of stating the most powerful version of rational
homotopy theory (Quillen’s, as far as I know) by baby steps.
(1) We’ll see how working rationally greatly simplifies many computations. In particular, computations with Eilenberg-MacLane spaces become easy, rather than a combinatorial nightmare producing Steenrod operations.
(2) In fact, we’ll show that if we are computing rationally, homotopy and homology don’t differ all
that much. This is achieved using Serre theory.
(3) Once this is achieved, we’ll discuss the rationalization of a space, and look at some specific examples, namely rational H-spaces. We’ll see that H-spaces become purely algebraic.
(4) I’ll review the construction of Postnikov towers, then note by earlier computations that these
should simplify drastically if we consider them rationally. This leads to a discussion of DGAs
and minimal DGAs.
(5) We will discuss Q-differential forms.
(6) We will discuss a version of the algebraic equivalence.
(7) We’ll get our grubby hands on some actual spaces, and do some computations.
(8) Finally, after the discussion of many algeraic gadgets, we’ll get to Quillen’s equivalence between
model categories
TopQ ↔ DGLAQ
So, that’s is nominally the goal of the course. The goal is also to introduce a number of topics of
independent interest. For example, some classical computations, simplicial sets, model categories, etc.
GrVectQ
O
U
proj
ChQ
O
1
Top≥
Q
/ CDAGQ
6
Top≥1
O
v
Set∆
2. Building Spaces
2.1. Postnikov Towers. Since we’ll be building a lot of spaces, it will be useful to discuss how they actually
get built. In particular, I’ll have to remind folks of Postnikov towers. Before I do that, I should remind
folks of building blocks of Postnikov towers, which are called Eilenberg-MacLane spaces.
2
Definition. An Eilenberg-MacLane space for a group G is a space K ( G, n) such that
(
πi K ( G, n) =
G
0
i=n
otherwise
Remark. One should note that for n = 1 the group may be non-abelian, but for n > 1, the group G must
be abelian, since it arising as the higher homotopy group of a topological space.
Example. S1 is a K (Z, 1). CP∞ is a K (Z, 2). RP∞ is a K (Z/2, 1).
One should maybe have an idea of why these exist.
Proposition. Let G be a group. Then a K ( G, 1) exists.
Proof Sketch. The idea is to create a fiber bundle G → EG → BG where EG is contractible. It will then
follow that π1 ( BG ) = G, and higher homotopy groups vanish. Thus, BG is a K ( G, 1). For a full proof, see
[?].
Proposition. Let G be an abelian group. Then K ( G, n) exists.
Proof. G will have a presentation
Z [r β ] → Z [ gα ] → G
where r β are the relations and gα are the generators. We consider α Sn — a bouquet of spheres with one
W
sphere for each generator. Every relation gives us a map ϕ β Sn → α Sn . Using these as attaching maps,
we form the pushout
W
W
β
W
/ Wα Sn
Sn
n +1
D
β
/X
We claim now that X has the property that πi ( X ) = 0 for i < n. We have a long exact sequence
πn+1 ( X, X (n) ) → πn ( X (n) ) → πn ( X ) → πn ( X, X (n) )
But, note that πn+1 ( X, X (n) ) = Z[r β ] and πn ( X (n) ) = Z[ gα ], πn ( X, X (n) ) = 0. So, the above exact sequence
actually gives a presentation.
Now, there are higher homotopy groups. But we go ahead and kill those off by inductively attaching
cells.
The following is (arguably) the most important property of Eilenberg-MacLane spaces, and it will be
used repeatedly and without much mention below.
Theorem. Let X be a space with the homotopy type of a CW-complex and let G be an abelian group. Then
H n ( X; G ) = [ X; K ( G, n)].
That is, cohomology is represented by K ( G, n).
We now note that we can build up topological spaces one homotopy group at a time by recourse to
Eilenberg-MacLane spaces.
3
Theorem. Let X be a path-connected topological space. A Postnikov tower is a diagram
···
X
G 3
> X2
/ X1
X
such that the following hold:
∼
=
• πi ( X ) −
→ πi ( Xn ) for i ≤ n.
• πi ( Xn ) = 0 when i > n.
• Xn → Xn−1 is a fibration with fiber K (πn ( X ), n).
I won’t prove that these things exist. They do. You can look it up in [1] or any other book on algebraic
topology. In fact, the provide successive approximations to a topological space in the following sense
Proposition. Let X be a connected CW-complex. Then the natural map
X → lim Xn
is a weak homotopy equivalence.
Given a random space X, it would be nice to sort of how to build the Postnikov tower. Suppose we’ve
started at the bottom with a map X → K (π1 ( X ), 1) that induces an isomorphism on π1 ( X ). We’ve like to
introduce a second space X2 such that we can produce a lift
K ( π2 ( X ), 2)
/7 X2
X
/ K ( π1 ( X ), 1)
(the K (π2 ( X ), 2) is hanging around to indicate that it is the fiber of the right vertical map). That is, we’d
like X2 to be a total space with fiber K (π2 ( X ), 2) and base K (π1 ( X ), 1). What we need are methods to
produce fiber spaces like this. One tried and true method of producing fiber spaces is by classifying maps
- in particular if we had a fibration sequence K (π2 ( X ), 2) → P → K (π2 ( X ), 3) where P is contractible, it
would be nice if we could extend the above to a diagram
K ( π2 ( X ), 2)
7/ X 2
/P
X
/ K ( π1 ( X ), 1)
/ K ( π2 ( X ), 3)
where the square on the left is now pullback. However, it is not always the case that we can do that. In
order for this to be the case, the putative X2 → K (π1 ( X ), 1) has to be a special kind of fibration, which
now define.
4
Definition. A fibration K (π, n) → E → B is said to be principal if it arises as a pullback from a path-loop
fibration. That is, if there is a diagram as below where the lower square is pullback:
K (π, n)
/ K (π, n)
E
/P
B
/ K (π, n + 1)
There is also a more general definition of principal:
Definition. A fibration F → E → B is a principal fibration if there is a commutative diagram
F
/E
/B
ΩB0
/ F0
/ E0
/ B0
such that the bottom row is a fibration sequence
Remark. Roughly, a fibration sequence is principal if it can be extended to the right, and not just the left.
We now have the following theorem, which I don’t think I will prove at the moment. It is pretty
standard and can be found in [1] or [2].
Theorem. A connected CW-complex has a Postnikov tower of pricnipal fibrations if and only if the action of π1 ( X )
on πn ( X ) is trivial.
Remark. Such a space is often called simple.
What is the utility of this? Given that there is a Postnikov tower with principal fibrations, we have at
each level of the postnikov tower a fibration sequence
k
n
X n +1 → X n −
→
K (πn+1 X, n + 2).
This means that Xn+1 is classified by an element
[ k n ] ∈ H n +2 ( X n , π n +1 X ).
These k-invariants, as they are known, often prove useful — as all classifying maps do.
Remark. A Look Ahead Why do we care about the above? Well, it’s a nice theoretical result of course.
But our goal here is to produce nice “rational” approximations to a space. It will turn out that it is very
easy to rationally approximate Eilenberg-MacLane spaces, and then we can bootstrap this to building up
rationalizations.
3. As Much as I Can Say About Spectral Sequences
SLOGAN: homology of the associated graded converges to the associated graded of homology.
For later use, it will behoove us to carefully discuss spectral sequences and where they come from. I
don’t know how much I’ll get to this in class.
Suppose there is some topological space you’d like to compute the homology of. This should beW the
case, given that you’re taking a topology class. If you’re lucky, the space splits up into parts X = i Xi
and you know how to take the homology of those parts. You are never this lucky. Instead, what usually
exists in nature is a filtration, i.e. some sequence of subspaces
∅ = X0 ⊂ X1 ⊂ X2 ⊂ · · ·
such that X = Xi . If there are only two steps in this filtration, i.e. X1 ⊂ X2 = X, you might hope to use
a long exact sequence in homology to get at H∗ ( X ):
S
· · · → Hi ( X1 ) → Hi ( X ) → Hi ( X, X1 ) → · · · .
5
You might again hope something like Hn ( X ) = Hn ( X1 ) ⊕ Hn ( X2 /X1 ). Of course, this is not true. What IS
true is that there is an induced filtration on homology:
p −1
p
Fn = Im( Hn ( X p ) → Hn ( X ))
Fn
p
⊂ Fn ⊂ · · ·
and that under right conditions we can compute the associated graded of H∗ ( X ):
p −1
M p
gr F Hn ( X ) =
Fn /Fn
Of course, this doesn’t completely recover homology (there are extension problems), but it gets us closer.
3.1. Two Degree Filtration. Let’s look at an example more carefully. In fact, disgustingly carefully.
Hn+1 ( X1 )
Hn+1 ( X2 )
Hn+1 ( X2 )
Hn+1 ( X2 )
/ Hn+1 ( X1 )
/0
/0
/0
/ Hn+1 ( X2 /X1 )
/ Hn ( X1 )
/ Hn ( X1 )
/0
/0
/ Hn ( X2 )
/ Hn ( X2 /X1 )
/ Hn−1 ( X1 )
/0
/ Hn ( X2 )
/0
/ Hn−1 ( X2 )
Let’s mess around with this a little bit. What we’ve decided is that we want to get at the associated graded
of homology. That is, we want
Im( H∗ ( X1 ) → H∗ ( X2 ))
H∗ ( X2 )/(Im H∗ ( X1 ) → H∗ ( X2 ))
We observe how these are contained in the diagram above.
First, we have
Im( Hn ( X1 ) → Hn ( X2 )) ∼
= Hn ( X1 )/ ker( Hn ( X1 ) → Hn ( X2 )
∼
= Hn ( X1 )/ Im( Hn+1 ( X2 /X1 ) → Hn ( X1 ))
And we also have
Hn ( X2 )/(Im Hn ( X1 ) → Hn ( X2 )) ∼
= Hn ( X2 )/(ker Hn ( X2 ) → Hn ( X2 /X1 ))
∼
= Im( Hn ( X2 ) → Hn ( X2 /X1 ))
∼
= ker( Hn ( X2 /X1 ) → Hn−1 ( X1 ))
If we make the definition
2
En,i
= ker( Hn ( Xi , Xi−1 ) → Hn−1 ( Xi−1 , Xi−2 ))/ Im( Hn+1 ( Xi+1 , Xi ) → Hn ( Xi , Xi−1 ))
we have a diagram
0
6
5 En+2,2
/ Im(0 → Hn ( X1 ))
6 En,1
Im( Hn+1 ( X1 ) → Hn+1 ( X2 ))
50
/ Im( Hn ( X1 ) → Hn ( X2 ))
6 En,2
0
0
/ Im( Hn ( X2 ) → Hn ( X2 ))
En2 +1,1
2
2
/0
2
/0
where the colored arrows indicate exact sequences (the paths parallel to the colored arrows are exact as
well).
6
3.2. Three Degree Filtration. We start with the diagram (which continues indefinitely in all directions):
/ Hn+1 ( X1 )
/0
/0
/0
/0
/ Hn+1 ( X2 /X1 )
/ Hn ( X1 )
/ Hn ( X1 )
/0
/0
/ Hn+1 ( X3 /X2 )
/ Hn ( X2 )
/ Hn ( X2 /X1 )
/ Hn−1 ( X1 )
/ Hn−1 ( X1 )
/0
/ Hn ( X3 )
/ Hn ( X3 /X2 )
/ Hn−1 ( X2 )
/ Hn−1 ( X2 /X1 )
/0
/ Hn ( X3 )
/0
/ Hn−1 ( X3 )
/ Hn−1 ( X3 /X2 )
Hn+1 ( X1 )
Hn+1 ( X2 )
Hn+1 ( X3 )
Hn+1 ( X3 )
Hn+1 ( X3 )
Now we turn the crank. For ease of notation, let
Imna,b := Im( Hn ( Xa ) → Hn ( Xb ))
2
En,i
= ker( Hn ( Xi , Xi−1 ) → Hn−1 ( Xi−1 , Xi−2 ))/ Im( Hn+1 ( Xi+1 , Xi ) → Hn ( Xi , Xi−1 ))
Then we have a diagram (where the colored arrows are exact sequences).
E2
/ Im0,1
En2 +1,1
/0
E2
/ Im1,2
En2 +1,2
/ Im0,1
n =0
0
/ Im2,3
En2 +1,3
/ Im1,2
n
0
/ Im3,3
<0
/ Im2,3
n
0
/ Im3,3
0
Im3,3
n
<
n +1
<
n +1
<
n +1
n +1
;
;
;
;0
/0
2
En,1
/0
2
En,2
/0
2
En,3
/ Im1,2
0
/ Im2,3
n +1
n −1
n −1
Note that we can prove these sequences are exact, just as we did above — by messing around with long
exact sequences. There is one exception, the sequence
3,3
2,3
2
2
1,2
0 → Im2,3
n+1 → Imn+1 → En+1,3 → Imn → Imn → En,2 → 0
For fun, we prove this sequence is exact (we at least do some of the argument). We focus on the following
portion of the diagram above:
Hn ( X1 )
i
Hn+1 ( X3 /X2 )
k
/ Hn ( X2 )
j
/ Hn ( X2 /X1 )
k
/ Hn−1 ( X1 )
i
Hn ( X3 )
2,3
2
0
First, let’s prove the map Im2,3
n → En,2 is surjective. Let x ∈ Imn . We know there is an x such that
2 (note that jx 0 is in the kernel of H ( X /X ) → H
ix 0 = x. We then take jx 0 . This is the map to En,2
n
2
1
n−1 ( X1 )).
7
Now, suppose
α ker( Hn ( X2 /X1 ) → Hn−1 ( X1 ))/ Im( Hn+1 ( X3 /X2 ) → Hn ( X2 /X1 ))
Thinking about it, this means α is in the image of j, i.e. there is α0 ∈ Hn ( X2 ) such that jα0 = α. Then look
at iα. This is in Im2,3
n . This shows the map is a surjection. (There is a little bit of a lie in this argument; see
if you can find it).
2
Now, we prove the sequence is exact at Im( Hn ( X2 ) → Hn ( X3 )). It is clear that the kernel of Im2,3
n → En,2
contains Im1,2
n (why?). But we need to show that it’s all of it.
2 . This means that
Let x ∈ Im( Hn ( X2 ) → Hn ( X3 )) and suppose that ix 0 = x and jx = 0 in En,2
0
0
jx ∈ Im( Hn+1 ( X3 /X2 ) → Hn ( X2 /X1 ). That is, there is a y such that jky = jx , i.e. j(ky − x 0 ) = 0. By
exactness, ky − x 0 ∈ Hn ( X2 ) lifts to an element in Hn ( X1 ), call it x 00 . Furthermore
i ( x 00 ) = i ( x 0 − ky) = ix 0 = x
since ik = 0 again by exactness.
I’ll stop proving exactness of this sequence here. The rest proceeds in substantially the same way. But!
The fact that we’ve verified exactness verifies that the diagram I drew above is actually useful. Now, we
can continue! But first, note that from the above diagram we have
2
En,1
= Im( Hn ( X1 ) → Hn ( X2 ))
1,3
2
= Im2,3
En,2
n / Imn
so that already at this stage we’ve aleady done some computing of the associated graded.
Define
3
3
En3 +1,i = ker( En,i
→ En3 −1,i−2 )/ Im( En3 +1,i+2 → En,i
)
We get a diagram
/0
En3 +1,1
0
0
E
0
En3 +1,2
/0
3
En,1
En3 +1,3
/0
3
En,2
0
Im1,3
n
0
Im2,3
n
0
Im3,3
n
E
Im1,3
n +1
E
Im3,3
n +1
0
F
Im2,3
n +1
F
3
En,3
F
0
0
We can now verify that the requisite sequences are exact. But note that they are short exact!
3.3. Exact Couples. The general mess of the above calls for some more general organizational system.
This is called an exact couple.
Definition. An exact couple is a diagram of (possibly graded) abelian groups
/D
i
D_
j
k
E
8

with the diagram exact at each corner.
Now, we define a map d : E → E by d = jk. Note that d2 = 0. We can then define new groups
E0 = ker d/ Im d
D0 = i( D)
and new maps as follows. First, let i0 = i | D . Let j0 (ix ) = [ jx ] (check that this is well-defined!). Let k0 [y] = ky
(again, check that this is well-defined!). These groups and maps give a derived couple ( E0 , D 0 , i0 , j0 , k0 ).
Proposition. The derived couple is an exact couple.
Proof. Diagram chasing exactly analogous to the above.
This means that the process can be iterated over and over again. In fact, what we were doing was
computing derived couples over and over again. We end up with a sequence ( Er , Dr , i(r) , j(r) , k(r) ) and we
define dr = j(r) k(r) . Now, a spectral sequence is a gadget that uses exact couples to compute (co)homology
from filtrations. The degree of cohomology and degree of filtrations give us a bigrading on our abelian
groups. That said, here is the definition of a spectral sequence.
Definition. A spectral sequence is a sequence { E∗r ,∗ , dr } of (possibly graded of bigraded) groups such
that
(1) dr : Er → Er satisfies d2r = 0
(2) Er+1 = ker dr / Im dr
Remark. The collection of groups E∗r ,∗ is typically called the rth page of a spectral sequence.
Remark. The differential dr has a degree. It’s degree depends on a few things; for example, whether we
are computing homology or cohomology. It depends on much more arbitrary choices, such as how we
index things. However, it is common in homological spectral sequences for dr to have degree (−r, r − 1)
and common in cohomological spectral sequences for dr to have degree (r, 1 − r ).
Example.
3.4. General Filtrations. Of course in this case, we have a buttload more information. We have exact
sequences like that for pairs ( Xi , Xi−1 ) and by direct summing the sequences we end up with a digram
like this:
L
/ L H∗ ( Xi )
H∗ ( Xi )
g
L
w
H∗ ( Xi , Xi−1 )
This gives us an exact couple,
3.5. Misc. There are other useful tools we’ll need from spectral sequences. For example, the following
comparison theorem.
r be two homology spectral sequences. Suppose that there is a map Er → E0r with
Theorem. Let Erp,q and E0p,q
p,q
p,q
∼ 0∞
E∞
p,q = E p,q
2 ∼ 02
E0,q
= E0,q
then
2
E2p,0 ∼
.
= E0p,0
Proof. ADD PROOF
Various exact sequences will also come in handy.
Example. Suppose we have a homological spectral sequence converging to some H∗ ( X ) with an E2 -page
such that
E2p,q = 0 0 < q < n.
9
That is, there is a large gap between the zeroth row and the next. The first differential that is interesting
2 . Since no other interesting differentials will hit the E -spot we have an exact
is then dn+1 : En2 +1,0 → E0,n
0,n
sequence
d n +1
2
∞
En2 +1,0 −−→ E0,n
→ E0,n
→0
Now, we further know we have a filtration of Hn ( X ):
Hn ( X ) = Fnn ⊃ Fnn−1 ⊃ · · · ⊃ Fn0 = 0
p
p −1
∞ = F0 . Since E∞ = 0 by assumption, we have F0 = F1 =
such that E∞
. In particular, E0,n
n
n
n
p,n− p = Fn /Fn
1,n
∞ . This continues until we hit
E0,n
∞
∞
∞
En,0
= Fnn /Fnn−1 = Fnn /E0,n
= Hn ( X )/E0,n
.
This gives an exact sequence
∞
∞
0 → E0,n
→ Hn ( X ) → En,0
→ 0.
Stitching these two exact sequences together we obtain
d n +1
∞
∞
2
→ E0,n
→ Hn ( X ) → En,0
→0
En2 +1,0 −−→ E0,n
4. The Serre Spectral Sequence
The Serre spectral sequence is a spectral sequence that will allow to organize information about fibrations. Given a fibration F → E → B it allows us to deduce information about H ∗ ( E) from H ∗ ( B), H ∗ ( F ).
Theorem (Homological Serre Spectral Sequence). Given a fibration F → E → B with B path connected and
π1 ( B) acting trivially on H∗ ( F, G ), there is a homological spectral sequence { Erp,q , dr } with
(1) dr : Erp,q → Erp−r,q+r−1 , i.e. a differential of degree (−r, r − 1)
∼ p p−1 where the F • arise from some filtration of H∗ ( E; G )
(2) E∞
•
p,n− p = Fn /Fn
(3) E2 ∼
= H p ( B; Hq ( F; G )).
p,q
Remark. In many cases, the E2 -page can be written as E2 ∼
= H p ( B) ⊗ Hq ( F ) by the Künneth theorem.
Remark. The picture to keep in mind:
•
•
•
•
•
•
•
•
•
•h
•
•
•
•
•
•
•
•j
• d4 •
•
•
•
•
•
•n
•
•
•
•
•
d3
•d2 •
•
•
There is also a cohomological version. This is perhaps more useful, since as wel’ll note (but not prove),
this comes equipped with a multiplicative structure that arises from the cup product in cohomology.
Theorem (Cohomological Serre Spectral Sequence). Given a fibration F → E → B with π1 ( B) acting trivially
p,q
on H ∗ ( B; G ), there is a cohomological spectral sequence { Er , dr } with
p,q
p,q
→ E p+r,q−r+1 with Er+1 = ker dr / Im dr
p,n− p ∼ n
(2) There is some filtration where E∞
= Fp /Fpn−1
(3) The E2 -page can be computed as
(1) dr : Er
p,q
E2 ∼
= H p ( B; H q ( F; G )).
This cohomology spectral sequence can be quite useful because of a multiplicative structure on it.
10
Theorem (Multiplicative properties of Serre Spectral Sequence). There is a multiplication
p+s,q+t
p,q
Er × Ers,t → Er
with the following properties
(1) dr is a derivation with respect to this product:
dr ( ab) = (dr a)b + (−1) p+q a(dr b)
(2) On the E2 -page the product is the cup product
H p ( B; H q ( F; R)) × H s ( B; H t ( F; R)) → H p+q ( B; H s+t ( F; R))
(3) The products on H ∗ ( E; R) restrict to products on the filtrations, and these agree with the product on E∞ .
Let’s do some examples.
Example. Let’s do a classic example: Compute H∗ (ΩSn ; Z). We use the fibration ΩSn → PSn → Sn . We
recall
(
Z ∗ = 0, n
n
H∗ (S ; Z) =
0 otherwise.
Now, the E2 -page is
(
E2p,q
n
n
= H p (S ; Hq (ΩS )) =
Hq (ΩSn ; Z)
0
p = 0, n
otherwise
So, let’s draw a picture in the case n = 3
H3 (ΩS3 ; Z)
0
0
H3 (ΩS3 ; Z)
0
0
H2 (ΩS3 ; Z)
H1 (ΩS3 ; Z)
0
0
H1 (ΩS3 ; Z)
Z
0
0
Z
0
1
2
3
H2 (ΩS3 ; Z)
j
The only differential that exists is d3 as pictured. Furthermore, since the E∞ page must be trivial, the map
must be an isomorphism. We have
H2 (ΩS3 ; Z) ∼
= H4 (ΩS3 ; Z) ∼
= H6 (ΩS3 ; Z) ∼
= ···
5. Some Rational Computations
We’ll use the Serre spectral sequence (mostly cohomological) to do some computations with coeffiecients
in Q.
The first few are perhaps the most important cases for us: The Eilenberg-MacLane spaces. As we’ve
seen in the construction of Postnikov towers, Eilenberg-MacLane spaces are the constituents of topological
spaces (in a sense dual to cells). Thus, knowing rational cohomology will give us a hint to the rational
cohomology of spaces in general.
Theorem. We have
(
∗
H (K (Z, n); Q) =
Q[ x ]
ΛQ ( x )
n even
n odd
where deg( x ) = n.
Proof. We first note that H ∗ (K (Z, 1); Q) ∼
= ΛQ ( x ) since S1 is a K (Z, 1). How, we note that there is the
path-space fibration
K (Z, n − 1) → PK (Z, n) → K (Z, n)
and then the Serre spectral sequence for this fibration has E2 term
p,q
E2 = H p (K (Z, n); Q) ⊗ H q (K (Z, n − 1); Q).
11
We are assuming by induction that we have computed H ∗ (K (Z, n − 1); Q). We have to work in cases.
Case 1: n odd. In this case H ∗ (K (Z, n − 1); Q) is a polynomial algebra on a generator degree n − 1.
First, we note that H ∗ (K (Z, n); Q) = 0 when ∗ < n and H n (K (Z, n); Q) = Qx. Let’s do n = 3. The E2 -page
is pictured as:
Qy2
0
0
Qx ⊗ Qy2
0
0
0
0
0
0
0
0
0
0
Qy
0
0
Qx ⊗ Qy
0
0
0
0
0
0
0
0
0
0
0
0
0
Qx
H 4 (K (Z, 3); Q)
H 5 (K (Z, 3); Q)
H 6 (K (Z, 3); Q)
&
&
Now, since the E∞ -page has to be zero, H 4 and H 5 both have to be zero, since there are no differentials
that hit it. The differential d2 doesn’t exist for degree reasons, so E2 = E3 . Now, the differential d3
(pictured) can be non-trivial. Since we need E∞ to vanish, it must be that dy = x. Then, this implies that
d(y2 ) = 2x ⊗ y and thus this is an isomorphism (since we’re working over the rationals). Since this must
be an isomorphism, H 6 has to vanish. Thus x2 = 0. We continue by induction to compute everything else.
Proposition. There is a map K (Z, 1) → K (Q, 1) induced by Z → Q descends to an isomorphism on cohomology:
∼
=
H ∗ (K (Z, n); Q) −
→ H ∗ (K (Q, n), Q)
6. Serre Theory
Definition. Let C be one of the following
• FG - finitely generated abelian groups
• T P - torsion abelian groups whose elements have orders that are multiplies of primes drawn from
P
• F P - the finite groups in T P .
There are some obvious consequences of the definitions which will be useful.
Lemma. We have
(1) The classes C are closed under extension. That is, if A ∈ C, B ∈ C then if there is an exact sequence
0→A→C→B→0
C ∈ C as well.
(2) If A, B ∈ C then A ⊗ B, Tor( A, B) ∈ C.
Exercise. Prove the above statement.
We are going after a kind of Hurewicz theorem mod C. The following lemma will be useful for that
Theorem. Let F → X → B be a fibration with π1 ( B) acting trivially the fiber. Assume F, X, B are path-connected.
If two of F, X, B have Hn ∈ C, then so does the third.
Proof. Suppose Hn ( F ), Hn ( B) ∈ C. We will figure out Hn ( X ) from the spectral sequence. The E2 page is
E2 = H p ( B, Hq ( F )) ∼
= H p ( B; Z) ⊗ Hq ( F; Z) ⊕ Tor( H p−1 ( B), Hq ( F )).
p,q
Every element in this E2 -page is in C by supposition. Since quotients of elements of C are in C, then Erp,q
and thus E∞
p,q ∈ C. So we have Hn ( X ) ∈ C.
There are two other cases, but these will be left as exercises to do on your own and look up.
Lemma. π ∈ C, then Hk (K (π, n); Z) ∈ C for all k, n > 0.
12
Proof. We will obviously use induction and the fibration sequence
K (π, n − 1) → ∗ → K (π, n).
There are some reduction steps.
• It suffices to do n = 1 by the previous lemma
• It suffices to do π = Z and π = Z/m by the Kunneth theorem (at least when C = FG or mb f Fp )
So, we do this for FG and F p . But, this is easy. A K (Z, 1) is S1 , and so the theorem holds. A K (Z/m, 1) is
a lens space, so this holds too.
Lemma. Let X be simply connected or π1 ( X ) acts trivially on πn ( X ), then
πn ( X ) ∈ C ⇐⇒ Hn ( X; Z) ∈ C
for all n > 0.
Proof. The conditions would see to suggest that we build a Postnikov tower. If πi ( X ) ∈ C, then H∗ (K (πi ( X ), n)) ∈
C from the above. Then, by using induction and the Serre spectral sequence, we get Hn ( Xn ; Z) ∈ C.
Before we go on, the following will be useful.
Lemma. Let X be a topological space with π1 ( X ) acting trivially on πm ( X ) for all m. Let Xn denote the n Postnikov
level of that space. Then
∼
=
Hn ( X ) −
→ Hn ( Xn ).
Proof. We begin with the map X → Xn and turn it into a fibration, which we also call X → Xn . Take the
fiber to obtain a fiber sequence
X > n → X → Xn .
Note that by the long exact sequence in homotopy, πi ( X ) = 0 for i ≤ n. Now, we apply the Serre Spectral
sequence. The E2 -page is
E2p,q = H p ( Xn ; Hq ( X >n ))
and we note that Hq ( X >n ) is only non-trivial for q ≥ n + 1. It follows that E∞
p,n− p = 0 except when p = n
and so Hn ( X ) = Hn ( Xn ).
Theorem (Mod C Hurewicz). If X has πi ( X ) ∈ C for i < n then h : πn ( X ) → Hn ( X ) is an isomorphism mod
C.
The following corollaries are usually more useful than the theorem itself:
Corollary. If for all i, πi ( X ) ∈ C, then h : π∗ ( X ) → H∗ ( X ) is an isomorphism mod C.
e i ( X; Z) ⊗ Q = 0 for all i if and only if πi ( X ) ⊗ Q = 0
Corollary. For X a simply connected topological space, H
for all i.
Proof of Theorem. We assume for the moment that X is simply connected. Let { Xi } be the Postnikov tower
of X. The maps πn ( X ) → Hn ( X ) and πn ( Xn ) → Hn ( Xn ) are the same, so we might as well work with the
latter. We of course have the fibration
K ( π n ( X ), n ) → X n → X n −1
and we can use our hammer of choice, the Serre spectral sequence. There is a 5 term exact sequence,
coming from the Serre spectral sequence:
Hn+1 ( Xn−1 )
d n +1
/ E∞
0,n
/ Hn (K (πn ( X ), n))
/ Hn ( Xn )
The composition of maps in the middle is the inclusion of the map
Hn (K (πn ( X ), n)) → Hn ( Xn ).
13
/ Hn ( Xn−1 )
Now, if we assume πi ( X ) ∈ C for i < n, then πn ( Xn−1 ) ∈ C and thus Hn ( Xn−1 ) ∈ C and Hn+1 ( Xn−1 ) ∈ C.
Thus, the map Hn (K (πn ( X ), n)) → Hn ( Xn ) is an isomorphism mod C. We then consider the diagram
π n ( K ( π n ( X ), n )
∼
=
∼
=
Hn (K (πn ( X ), n)
/ π n ( Xn )
/ Hn ( Xn )
The left vertical map is an isomorphism by regular Hurewicz. The top horizontal map is an isomorphism
by definition. We just showed the bottom horiztonal map is an isomorphism mod C. Thus, the right
vertical map is an isomorphism mod C.
7. Rational Homotopy Groups of Spheres
We now have enough information to prove an actual, big theorem in algebraic topology: we get some
structure results on the homotopy groups of spheres.
Theorem. πi (Sn ) are finite for i > n except for when n is even and i = 2n − 1. In this case
π2n−1 (Sn ) = Z ⊕ finite.
Proof. Consider the map Sn → K (Z, n) which represents the generator of πn (K (Z, n)) ∼
= Z. We can
consider this map to be a fibration with fiber F. By the long exact sequence in homotopy for a fibration
πi F ∼
= πi Sn in the range we care about (i > n). Further convert the map F → Sn into a fibration with fiber
K (Z, n − 1):
K (Z, n − 1) → F → Sn .
This is amenable to attack by the Serre spectral sequence. Since K (Z, n − 1) have different homology
groups when n is even or odd, we split into two cases.
Case 1: n odd. In this case we know the E2 -page is
E2p,q = H ∗ (Sn ; Q) ⊗ H ∗ (K (Z, n − 1); Q) ∼
= Λ( x ) ⊗ Q[y]
and so it looks like (with d2 drawn in)
3n − 3
Qy3
2n − 2
Qy2
n−1
Qy
0
Q
0
Qxy3
&
Qxy2
&
&
Qxy
Qx
n
Now, we know that F is (n − 1)-connected, so if we are to kill off any cohomology in dimension n − 1, we
must have that d2 : Qy → Qx is an isomorphism. By the multiplicativity of the spectral sequence it is then
e ∗ ( X; Q) = 0, and πi ( X ) ⊗ Q is 0 as well. Thus, πi (Sn ) is
the case that ALL d2 ’s are isomorphisms. Thus, H
finite for i > n.
Case 2: n even. In this case the E2 -page is
E2p,q = H ∗ (Sn ; Q) ⊗ H ∗ (K (Z, n − 1); Q) ∼
= Λ( x ) ⊗ Λ(y)
14
and so we have the E2 -page looking like
Qy
Qxy
"
Q
Qx
e ∗ ( X; Q) ∼
e (S2n−1 ; Q). By Hurewicz for C = FG, we have that πi (Sn ) is 0 for
which means that H
= H
n
n < i < 2n − 1 and π2n−1 (S ) = Z ⊕ finite.
We now need to compute the higher homotopy groups. Let F → F <2n−1 be a map obtained by killing
all homotopy groups i ≥ 2n − 1 of F. We turn that map into a fibration to obtain a fiber sequence
F ≥2n−1 → F → F <2n−1 .
e ( F <2n−1 ; Q) =
We note that all πi ( F <2n−1 ) are finite (since we already know them for i < 2n − 1), and thus H
0. The Serre spectral sequence then gives
H ∗ ( F ≥2n−1 ; Q) ∼
= H ∗ ( F; Q) ∼
= H ∗ (S2n−1 ; Q).
We now consider
F ≥2n−1 → K (Z, 2n − 1)
which induces an isomorphism on π2n−1 mod torsion. We use the argument as in the odd case to get that
πi ( F ≥2n−1 ) is finite for i > 2n − 1. That is, we extend to a fibration sequence
K (Z, 2n − 2) → Fe → F ≥2n−1 → K (Z, 2n − 1).
We have, as in our earlier case πi ( Fe) = πi ( F ≥2n−1 ) for i > 2n − 1, so it’s enough to compute homotopy
e How, we use the Serre spectral sequence. The E2 -page is
groups of F.
E2p,q = H p ( F ≥2n−1 ; Q) ⊗ H q (K (Z, 2n − 2); Q) = H p (S2n−1 ; Q) ⊗ H q (K (Z, 2n − 2); Q).
e Q) has to vanish.
We then argue as in the first case to show that H i ( F;
8. Rational Spaces and Localization
We construct rationalizations of spaces. That is, for a topological space, there is a functorial construction
which assigns to X a new spaces XQ and a map X → XQ such that π∗ ( X ) ⊗ Q → π∗ ( XQ ) ⊗ Q = π∗ ( XQ )
is an isomorphism. In fact, more is true about localization. Let’s precisely state what a Q-local space is
Definition. A Q-local space (or a Q-space) is a simply connected topologocal space X such that
(1) π∗ ( X ) is a Q-vector space
e ∗ ( X; Z) is a Q-vector space.
(2) H
Note that this is implicitly a theorem too — we have to prove that those two conditions are equivalent
Proof. 1) =⇒ 2). We assume that πi ( X ) is a Q-vector space. To show that Hi ( X ) is a Q-vector space,
we use our favorite tool: Postnikov towers and the Serre spectral sequence. The induction can start at
K (π2 ( X ); Q) = X2 . Then we know that H∗ ( X2 ; Z) is a Q-vector space if π2 ( X ) is. We assume that we’ve
proved that H∗ ( Xn−1 ) is a Q-vector space. We use the fibration
K ( π n ( X ), n )
/ Xn
X n −1
The En -page of this spectral sequence is
E2p,q = H p ( Xn−1 ; Z) ⊗ Hq (K (πn ( X ), n); Z)
then this gives that H∗ ( Xn ; Z) must be a Q-vector space.
15
e ∗ ( X; Z) is a Q-vector space. And suppose by induction that we know π∗ ( Xn−1 ) are
2) =⇒ 1) Suppose H
Q-vector spaces. Then H∗ ( Xn−1 ; Z) is (by the above) and so is H∗ ( X, Xn−1 ) (by the long exact sequence of
a triple. But πn ( X ) = Hn+1 ( X, Xn−1 ). So, all homotopy groups of Xn are rational vector spaces.
Say something about commuting with limits
Definition. Let X, XQ be simply-connected CW-complexes. We call a map X → XQ a localization if any
of the three (equivalent!) conditions are satisfied:
(1) The induced map
e ∗ ( X; Q) → H
e ∗ ( XQ ; Q )
H
is an isomorphism.
(2) The map
π ∗ ( X ) ⊗ Q → π ∗ ( XQ ) ⊗ Q
is an isomorphism
(3) The map X → XQ is universal in the following sense. If X → YQ is a map to a Q-space YQ then
this factors through XQ uniquely.
Proposition. The above conditions are equivalent.
Proof. I won’t prove all of them (one of the equivalences requires an argument from obstruction theory,
which I would prefer to ignore for the moment).
3) =⇒ 1) Let YQ = K (Q, n). Then the fact that this factors uniquely means that
∼
=
[ X, K (Q, n)] ←
− [ XQ , K (Q, n)]
and thus H ∗ ( X; Q) ∼
= H ∗ ( XQ ; Q) by the representability theorem.
2) ⇐⇒ 1) In this case π∗ ( X ) ⊗ Q → π∗ (Y ) ⊗ Q will be an isomorphism if and ony if π∗ ( F ) ⊗ Q = 0 for
all i. But, by the rational forms of Hurewicz, this will be the case if and only if H∗ ( F; Q) = 0. But by the
Serre spectral sequence, this will imply that H∗ ( X; Q) ∼
= H∗ (Y; Q).
We would now like to construct localizations of spaces. That is, we would like to construct a map
X → XQ such that XQ is Q-local and the map induces an isomorphism on rational cohomology.
First, we note that for K ( G, n) with G abelian, this is easy.
Proposition. The map K ( G, n) → K ( G ⊗ Q, n) induced by G → G ⊗ Q is a Q-localization.
Proof. By definition.
Using K ( G, n) as building blocks, we can construct the localization of a space in general. Note that the
above only makes sense when G is abelian and so for the moment we will neglect π1 phenomena.
Theorem (Q-localization of spaces). Given a (for now simply connected) CW-complex X, there is a localization
XQ and a map X → XQ .
Proof. We construct this via inducation and Postnikov towers. Suppose we have constructed a localication
`n−1 : Xn−1 → ( Xn−1 )Q . We’d like to construct ( Xn )Q . Well, we know how to construct Xn — via the
k-invariant. That is to say, there is a (pullback) commutative diagram
/∗
Xn
X n −1
K ( π n ( X ), n + 1).
kn/+1
We know exactly what we want the n-level of ( Xn )Q to be — we’d like it to be K (πn ( X ) ⊗ Q, n + 1).
We can compose the k-invariant above with the localization map:
Xn−1 → K (πn ( X ), n + 1) → K (πn ( X ) ⊗ Q, n + 1)
16
and this, by definitino of localization factors through ( Xn−1 )Q . We now form the pullback square to define
( Xn ) Q
( Xn ) Q
/∗
( X n −1 ) Q
/ K (πn ( X ) ⊗ Q, n + 1)
and there is an induced map Xn → ( Xn )Q . It is easy to check that this is the desired localization.
Example (Rational Bott Periodicity). Chern classes produce maps (by representability)
BU →
∏ K(Q, 2k)
k
and the computation of
Thus,
H ∗ ( BU; Q)
show that that map becomes an isomorphism on rational cohomology.
BUQ '
∏ K(Q, 2k).
k
In particular, we note Ω2 BUQ ' BUQ , which is rational Bott periodicity.
The following should properly be an example, but it’s interesting enough to state as a proposition.
2k +1
Proposition. SQ
is a K (Q, 2k + 1). Also, there is a fibration
2k
K (Q, 4k − 1) → SQ
→ K (Q, 2k)
so that
(
2k
πi ( S ) ⊗ Q =
Q
0
i = 2k, 4k − 1
otherwise
Proof. We know that S2k+1 → K (Q, 2k + 1) induces an isomorphism on rational cohomology (thus homology, thus homotopy). Or, we can observe that K ( Q, 2k + 1) is a Moore space M (Q, 2k + 1), but
2k +1
M(Q, 2k + 1) = SQ
.
For the rest, consider
2k
SQ
→ K (Q, 2k)
which induces an isomorphism on H2k (we can get such a map from the universal property of localization).
Turn this into a fibration
2k
F → SQ
→ K (Q, 2k)
and use the Serre spectral sequence
Q
Qx
Qx2
Qx3
0
2k
4k
6k
8.1. Rational H-spaces.
17
9. DGAs
We now switch gears. We’ve discussed how to build spaces, and how to do some computations. We’ve
discovered that if we are working with only rational coefficients, computations become much easier. There
is, of course, a deeper reason for this. We will see later that when we put on our rational goggles,
topological spaces are classified completely by certain types of DGAs. Furthermore, notions of maps and
homotopies will extend to the category of DGAs. Since this is the case, it is important for us to understand
the homotopy theory of DGAs.
9.1. Basic Definitions.
Definition. A differential graded algebra is an algebra that is both graded, and equipped with a differential and the algebra multiplication and differential play well with the grading. More precisely, A∗ is an
algebra over a field k (in our case, Q, usually) with a grading A p such that
A∗ =
M
Ap
p ≥0
and
(1) A differential
d : A∗ → A∗+1
with d2 = 0 and
d( ab) = da · b + (−1) p a · db
(2) The multiplication is graded: A p ⊗ Aq → A p+q satisfies
ab = (−1) pq ba.
What are some examples?
Example. —
(1) Over R, differential forms under wedge product.
(2) (t, dt). This is going to be the “interval” in much of what follows. This is kind of lame, not-too-clear
notation for the algebra
Q[ x ] ⊗ Λ(y) dx = y, deg x = 1, deg y = 2.
Let’s just do a computation with this for fun:
d( xy) = xdy + (−1)0 ydx = xd2 x = 0
d( x n y) = x n dy + (−1)n d( x n )y = (−1)n n!(dx )y = 0
We can even compute the cohomology! And we will, for shits and giggles. We have
H 0 ( I ) = ker d : I 0 → I 1 = Q
since the differential operating on the ground field is 0. Moving on,
H 1 ( I ) = ker d/ Im d = 0
since dx = y and all the elements in degree 1 are of the form ax.
H 2 ( I ) = Qhyi/Qhyi
(3) The Koszul complex
(4) H ∗ ( X; Q) with a trivial differential (i.e. it’s a graded vector space with a multiplication)
(5) H ∗ ( X; Z) with Bockstein.
Example. An important non-example is C ∗ ( X; Q).
Example. There isn’t really a notion of free DGA (as far as I know), but there are semi-free DGAs. These
are DGAs whose underlying graded-commutative algebra is free. Free graded-commutative algebras are
defined as follows. Given a graded vector space {V ∗ } the free graded-commutative algebra is
F (V ∗ ) = k[Veven ] ⊗k Λ(Vodd ).
18
Definition. The cohomology of a DGA is defined how one would think it is. There is a differential
d : A∗ → A∗+1 and we define
H n ( A∗ ) = kerd : An → An+1 / Im d : An−1 → An
Definition (Relative Cohomology). Let C ∗ , D ∗ be chain complexes (we forget structure from DGAs), and
suppose f : C ∗ → D ∗ is a map between chain complexes. We define a mapping cylinder, M f to be
Mnf = C n ⊕ D n−1
with differential d M : Mnf → Mnf +1 given by
We then make the definition
dC
f
0
−dC
H ∗ (C, D ) := H ∗ ( M f ).
Lemma. There is a log exact sequence
→ H n ( C ∗ ) → H n ( D ∗ ) → H n +1 ( C ∗ , D ∗ ) → H n +1 ( C ∗ ) → · · ·
Proof. We leave this as an exercise.
9.2. Sullivan Models. We would now like to do homotopy theory with DGAs. We could wave a wand
and come up with a model category structure on CDGAs. And we will. But it’s also useful to know how
to work with them on an element-by-element level, so we’ll develop the abstract and concrete in parallel
(also as a good example of the abstractions).
Recall we have the following model structure on CDGAQ
Theorem. There is a model structure on CDGAQ such that
• The generating cofibrations, I, are the maps ΛSn−1 → ΛD n
• The generating ayclic cofibrations, J, are the maps 0 → ΛD n .
• The weak equivalences are quasi-isomorphisms.
Furthermore, the cofibrations are retracts of I-cell complexes and the fibrations are (degree-wise) surjections.
Now, this is a cofibrantly generated model category. This allows us to quickly identify the cofibrant
objects in this category (or at least some of them).
Corollary. The cofibrant objects in CDGAQ with the above model structure are exactly I-cells.
A convenient property:
Corollary. All objects in CDGAQ are fibrant.
Definition. A Sullivan algebra is a cofibrant object in CDGAQ .
Example. Let’s see what this says. Suppose we’ve built k − 1 stages of a Sullivan algebra, and we call the
algebra ΛV (k − 1) at stage k − 1. To build the next stage we build a pushout
ä ΛSn
/ ΛV (k − 1)
ä ΛD n
/ ΛV (k)
The map ΛSn → ΛV (k − 1) picks out an x ∈ V (k − 1) and the pushout forces the condition that dx = y,
where y corresponds to the element that D n maps to in ΛV (k).
Remark. Look up the definition of a Sullivan algebra in Felix-Halperin-Thomas. The definition they have
there is completely equivalent to this one, but this one is a bit more compactly stated.
Remark. A lot of times below we’ll suppress the notation of the derivation.
19
Definition. A Sullivan model for A ∈ CDGAQ is a sullivan algebra ΛV such that there is a quasiisomorphism ΛV → A.
Now, one can easily see from our knowledge of model categories that a Sullivan model exists — this is
because a Sullivan model is just cofibrant replacement!
We now want to define homotopies. We need a quick auxilliary definition
Definition. Λ(t, dt) = ΛD1 .
Definition. Two morphisms f , g : A → B of CDGAQ are homotopic if there is a morphism
H : A → B ⊗ Λ(t, dt)
such that (Id ⊗e0 ) H = f and (Id ⊗e1 ) H = g.
Remark. Of course, it is easy to check that B ⊗ Λ(t, dt) is a cylinder object for B and so the definition
above corresponds with our usual definition of homotopy in a model category.
We have a bunch of lifting properties for free from the model structure. They are usually a pain to
prove because people insist on using algebraic structure. For some of the proofs we’ll do that too, just to
get a sense of how to do element-wise arguments in CDGAQ .
Lemma (The Lifting Lemma). Suppose we are given a diagram
=A
'
/B
ΛV
where the right vertical arrow is a surjective quasi-isomorphism and ΛV is a sullivan algebra. Then there is an
indicated lift.
Proof 1. A surjective quasi-isomorphism is a trivial fibration. Since ΛV is cofibrant, the lift exists.
Proof 2. ΛV can be inductively built, and so, as usual for cellular things, we build a lift inductively.
Suppose we’ve lifted ΛV (k − 1). The next level is determined by a map d : Vk → Λ(k − 1). Let Vk =
h v α i.
Definition. A relative Sullivan algebra is a relative I-cell complex.
Remark. As usual, this has an alternate definition. It is a CDGAQ of the form B ⊗ ΛV where B is a
subcochain algebra, 1 ⊗ V = V and V is built as V (0) ⊂ V (1) ⊂ · · · such that
d : V (0) → B
d : V (k) → B ⊗ ΛV (k − 1)
Lemma. B → B ⊗ ΛV is a cofibration.
Proof. It’s obviously a relative I-cell complex.
One would expect that we’d be able to factor maps into these guys. This is, after, all exactly what the
small object argument did for us. Let’s first give a quick definition
Definition. Let A ∈ CDGAQ . Let U A denote the underlying graded vector space
E( A) = Λ(U A ⊕ dU A)
∼
=
d : UA −
→ dU A
The literature of rational homotopy gives the following a special name (the “surjective trick”) but it’s
really factorization in a model category.
Theorem. Let A → B be a morphism in CDGAQ as
'
A−
→ A ⊗ E( B) → B
where the first map is a relative Sullivan algebra and E( B) → B is surjective.
20
Proof. Actually quite easy to see from definitions, but also follows from factorization.
We continue to develop homotpy theory.
Definition. Let [ A, B] denote the set of homotopy classes of maps of CDGAQ ’s from A to B.
Proposition. Let A → C be a quasi-isomorphism and ΛV a Sullivan algebra. Then
[ΛV, A] → [ΛV, C ]
is a bijection.
Proof. A and C are both fibrant objects, and ΛV is cofibrant. This then follows from a statement about
model categories. This is one of those cases where doing an elemental analysis of it just isn’t worth it. Lemma. Given a diagram
/
;A
B
L
f '
/C
B ⊗ ΛV
ψ
where the left vertical arrow is a relative Sullivan model and the right vertical arrow is a quasi-isomorphism there is
a lifting L such that f ◦ L ' ψ.
It is high time for an example.
Example. Well, a classic non-example. Consider the algebra
A = Λ( a, bc)
deg a, b, c = 1
where
da = bc
db = ac
dc = ab
This is not a Sullivan algebra. Why?
However, it can be approximated by a sullivan algebra. The whole algebra can written as
0 → h a, b, ci → h ab, bc, cai → h abci → 0 → 0 → · · ·
Computing homology is easy. ker d : A1 → A2 = 0, so H 1 ( A) = 0. ker d : A2 → A3 = h ab, bc, aci which is
the whole image of d : A1 → A2 , so H 2 ( A) = 0 .Finally, d : A3 → A4 is abc, which is not in the image of
d : A2 → A3 . Thus H 3 ( A) = Qh abci. Define a CDGA Λ( x ) with deg x = 3 and no differential. Then it is
easy to check that there is a quasi-isomorphism
f : Λ( x ) → A
f ( x ) = abc.
9.3. Minimal Sullivan Models. Sullivan algebras are I-cell CDGAs, which should be thought of like
cellular topological spaces. However, we all know from working with topological spaces that the concept
of “CW” is also very important. The version of this for CDGAs is called a “minimal Sullivan algebra.”
These, like CW complexes, will be easier to work with and have a sort of uniquness that Sullivan algebras
lack.
Definition. Let Λ+ V denote the positive-degree elements of ΛV in positive degree.
Definition. A Sullivan algbra is a minimal Sullivan if
Im d ⊂ Λ+ V · Λ+ V.
Lemma. Any CDGA of the form (ΛV, d) with V = V ≥2 such that Im d ⊂ Λ+ V · Λ+ V
21
Proof. It already satisfies the condition for minimality. Consider the degree k chunk of V, V k . We know
d : V k → V k+1 , but also Im d ⊂ Λ+ V · Λ+ V. Looking at (Λ+ V · Λ+ V )k+1 we see for degree reasons that it
is inside ΛV ≤k−1 . Thus, d : V k → ΛV ≤k−1 , which provides us with the filtration necessary for a Sullivan
algebra.
We know that every algebra has a Sullivan algebra replacement (since it is just cofibrant replacement).
We can’t use this technique to produce minimal Sullivan algebras. We’ve got to do this by hand. We’ll
provide a general argument later, but for simply connected algebra, we really can do an element by element
argument.
Proposition. Suppose A ∈ CDGAQ such that H 0 ( A) = Q and H 1 ( A) = 0. Then there is a minimal Sullivan
model (ΛV, d) for A.
Proof. We, of course, proceed by induction. We start H 2 ( A). Just pick ΛV 2 with no differential and a map
f 2 : (ΛV 2 , 0) → A. We suppose that f k : (ΛV ≤k , d) → A has been constructed and we try to construct
f k+1 : ΛV k+1 → A.
We also suppose by H i ( f k ) is an isomorphism for i ≤ k and injective for i = k + 1. We try to construct
f k +1 .
Consider the long exact sequence
H k +1 ( f k )
H k+1 (ΛV ≤k )
H ks +2 ( f k )
H k+2 (ΛV ≤2 )
/ H k+1 (ΛV ≤k , A)
/ H k +1 ( A )
/ H k +2 ( A )
We assume that H k+1 ( f k ) is injective. We’d like to make it surjective as well, so we’ll try to tack on some
extra stuff to ΛV ≤k to make this true. Similarly, we’ll try to fix up H k+2 ( f k ) to be injective.
We work on the first part. A failure of surjectivity is that H k+1 ( f k ) “doesn’t hit enought stuff.” We pick
a basis for the stuff it doesn’t hit, i.e.
H k+1 ( A) = Im H k+1 ( f k ) ⊕
M
Q h ai i
i∈ I
where the ai is the basis and I is some indexing set. Also, a failure of injectivity means we have a kernel
for H k+2 ( f k ). We pick a basis for this too:
ker H k+2 ( f k ) =
M
Q h z j i.
j∈ J
So at least we have names for what’s mucking us up. Now, we build these things into our algebra. Let
V k+1 = hvi0 , v00j i I,J
where deg v0 , v00 = k + 1 and define
dvi0 = 0
dv00j = z j .
This makes sure that vi0 represent elements of homology in degree k + 1 and makes sure that z j is exact.
We let ΛV ≤k+1 = ΛV k+1 ⊗ ΛV ≤k . Now, we extend the map to f k+1 : ΛV ≤k+1 → A by
f k+1 (vi0 ) = ai
f k+1 (v00j ) = b j
where f k z j = db j (this is just to make sure the differential works).
Now we show that the inductive hypothesis holds.
H k+1 ( f k+1 ) is surjectivey by construction. We now want both H k+1 ( f k ) and H k+2 ( f k+1 ) to be injective.
Suppose f k+1 [α] = 0 where α is in degree k + 1. That is, we can write a representative
α=
∑ λi vi0 + ∑ λ j v00j + R
where R is a product of lower order terms.
22
Since α is a closed thing, dα = 0. Thus
∑ λ j z j + dR = 0.
In homology, this becomes ∑ λ j [z j ] = 0, and because we chose z j to be a basis, each λ j = 0. So, now we
know α = ∑ λi vi0 + R. Apply f to get
f ( α ) = ∑ λi αi + f ( R )
Now, in homology this is ∑ λi [αi ] + [ f ( R)]. However, we know that [ f ( R)] ∈ Im H k+1 ( f k ) and [αi ] is a
basis for the rest. We thus have λi = 0.
We need to show that H k+2 ( f k+1 ) is injective now. But this is by construction. We’ve made everything
in ker H k+2 ( f k ) exact, and thus 0 in homology.
9.4. Sullivan and Minimal Models. We are going to claim that DGAs over Q provide a reasonable model
for rational spaces. Spaces, as we know, can be built out of smaller bits in various ways. For example, we
can build a cell complex or we can build a Postnikov tower (these are in a precise way dual to each other).
In the same way, and perhaps it is even simpler to see, various algebra things are built out of smaller
bits. Vector spaces have bases, algebras have generators and relations and these objects can be built up
sequentially out of such things.
Definition. A DGA A∗ is minimal if
(1) A∗ is free as a graded-commutative algebra on generators in degree ≥ 2.
(2) We also have
M
d( A∗ ) = A+ ∧ A+
A+ =
Ak
k >0
We now define how to build DGAs as we build Postnikov towers.
Definition. Let A∗ be a DGA. A Hirsch extension is an inclusion
A ∗ → A ∗ ⊗ d Λ (V k ).
The notation on the right means that we come equipped with a map d : V → Ak+1 .
We say two Hirsch extension are equivalent if...there is an obvious commutative diagram.
Lemma. Two extensions
A → A ⊗ d Λ (V )
A → A ⊗ d0 Λ (V 0 )
are equivalent iff there is an isomorphism ψ : V → V 0 and the diagram commutes:
/ V0
V
#
{
H k +1 (A )
∼
=
Proof. =⇒ direction. We assume that ϕ : A∗ ⊗d Λ(V ) −
→ A∗ ⊗d Λ(V 0 ) is an isomorphism and that
∗
it furthermore extends the identity on A . Because of the condition of extending the identity, and the
diagram commuting, it must be the case that
ϕ(v) = av + ψ(v)
where av is an element of A∗ depending on v (linearly). Now,
ϕ(dv) = d0 ( ϕ(v)) = d0 ( av + ψ(v)) = d0 ( av ) + d(ψ(v)).
This means that
[dv] = [d0 ψ(v)] ∈ H k+1 (A∗ ).
23
∼
=
→ V 0 then [dv] = [d0 ψ(v)] ∈ H k+1 ( A∗ ). This means
⇐= direction If ψ : V −
dv − d0 ψ(v) = da, av ∈ A∗ .
Then, define ϕ(v) = av + ψ(v) and this defines a map.
The corollary of the proof is more important than the proof itself:
Corollary. Equivalence classes of extensions are in 1-1 correspondence with maps d : V → H k+1 ( A∗ ) which in
turn is the same thing as a class in H k+1 ( A∗ ; V ∗ ).
The following gives a characterization of minimality, and tells us why we introduced Hirsch extensions
in the first place.
Theorem. Let M be a DGA and M(n) ⊂ M the subalgebra generated in degrees ≤ n. Then M is minimal if
S
M(1) is the ground field, n M(n) = M and each inclusion M(n) ⊂ M(n + 1) is a Hirsch extension with the new
generator in degree n + 1.
10. Forms on Simplicial Sets
We’ve studied rational spaces and dgas. We’re going to try to link them together now. To a space, we’ll
want to assign a certain DGA. The issue, of course, is what DGA to assign to it.
A first guess would be the chain complex C ∗ ( X; k ) , but chain complexes are a somewhat poor proxy
for our purposes. The issue with them is that there is only a quasiisomorphism between C ∗ ( X × Y; k )
and C ∗ ( X; k) ⊗k C ∗ (Y; k ). If X were a manifold we’d have a ready-made DGA: the complex of differential
forms Ω∗ ( X; R) under wedge product.
The idea is to create differential forms for more general types of spaces, e.g. simplicial sets. Of course,
differential forms have their own theory which must be developed.
10.1. Differential Forms. We recall the following:
Definition. Let C be a category. A simplicial object in C is a functor X : ∆op → C
Remark. Simplicial rings arise in derived algebraic geometry, simplicial abelian groups pop up all over
the place (e.g. higher Chow groups), simplicial simplicial sets (alias bisimplicial sets) are quite useful, etc.
Definition. A simplicial commutative DGA is a functor A : ∆op → CDGAQ,∆ .
Remark. Note that a simplicial CDGA has two indices, and we will often denote one by A•• . The upper
index will be the algebra grading and the lower index will be the simplicial grading.
p
Remark. We think of A• as a simplicial set (by forgetting structure)
We come to a very important definition
Definition. Let A•• ∈ CDGAQ,∆ and K ∈ Set∆ . Define A p (K ) as follows. As a set
p
A p (K ) = HomSet∆ (K, A• ).
Addition, multiplication, and the differential are given by the second coordinate. We note that this construction is covariant in the algebra and contravariant in the simplicial set.
Remark. This is slightly abstract. What is this saying? For every simplex σ ∈ Kn (i.e. every map ∆n → K)
p
we assign it an element xσ ∈ An . That is, Φ ∈ A p (K ) is a map Φ(σ ) = xσ . The map is additive, plays well
with scalar multiplication and the differential.
Remark. A further remark: How should we think of this in relation to differential forms on manifolds?
Why doesn’t the definition look the same? First, recall the definition of Ω p ( M ). We first define a bundle
Λ p ( T ∗ M ) → M and then define Ω p ( M ) to be a section of this bundle. A section, of course, is a continuously varying choice of p-form. That is, it’s a continuously varying choice of some symbol that transforms
correctly between charts. So what does it take to define a p-form? Really, a way to define it on a chart
(local data), and then a way to move between charts (gluing).
24
What does this look like for simplicial sets? The local data for simplicial sets is simplices and the gluing
data is basically face and degeneracy maps. So, to each simplex we assign a formal p-form and then we
glue along faces via face maps. How do we assign to each simplex a p-form? Just have a simplicial map
p
p
from K to A• . So, in some sense A• (which is a simplicial set) can be thought of as a classifying space for
p-forms.
In fact, this is how “classifying spaces” work in ∞-category theory.
Corollary. Let A•• ∈ CDGAQ,∆ . Then
∼
=
→ An .
A(∆n ) −
The idea will be to use these constructions to make differential forms. We would like to construct
differential forms on ∆n first. What should these look like? Let’s think of ∆n as an actual topological space
— in this case it corresponds to the set ∑ ti = 1. We would thus expect differential forms to be things that
look like ∑ f (t0 , . . . , tn )dt0 ∧ · · · ∧ dtn (subject to the relations imposed by ∑ ti = 1). We need to turn this
into an actual simplicial construction.
Definition. We define an algebra (the n-forms)
( A PL )n = Λ(t0 , . . . , tn , dt0 , . . . , dtn )
∑ ti − 1, ∑ dti
And we can further turn ( A PL )• into a simplicial object by defining face maps


k<i
tk
di ( t k ) = 0
k=i


t k −1 k > i
and degeneracy maps


tk
s j ( t k ) = t k + t k +1


t k +1
k<j
k=j
k>j
We are ready to make our main definition here. This is really the main point of the course. We’ll be
proving theorems about the following object for a little while.
Definition (? ? ?). Let K ∈ Set∆ . We define
A PL (K ) = ( A PL )•• (K ).
Note that A PL (K ) is in CDGAQ,∆ .
Remark. This is a way of associating to a “space” to some commutative differential graded algebra. This
is what we have been seeking all along. In general, to give away the game, what we’ll do is associate to a
“space” K, a minimal sullivan model ΛV → A PL (K ). This, by uniqueness of Sullivan algebras, will be a
good representative for K.
Corollary. A PL (∆n ) = ( A PL )n
10.2. de Rham Theorem and Poincare Lemma. We would now like to show that the construction above
is cohomologically well-behaved. This is the point of differential forms after all. The game here is going to
be mimicking certain constructions and theorems from the theory of smooth R-valued differential forms.
In particular
(1) The Extension Lemma (Poincare’s lemma for star shaped domains)
(2) Poincare’s Lemma
(3) the de Rham theorem, i.e. relating Ω∗ ( X; R) to C ∗ ( X; R).
Let’s recall the main theorems.
Theorem (The Poincare’s Lemma). Let U be a contractible (or star-shaped) open subset of Rn . Let ϕ ∈ Ω(U; R).
Suppose dω = 0, then there is an α such that dα = ω.
25
Theorem (The de Rham Theorem). There is a map Ω∗ ( X; R) → C ∗ ( X; R) given by
ω 7→
Z
(−)
ω
induces an isomorphism on cohomology.
The goal will be to prove these in the case of rational differential forms. We begin with the extension
lemma. This is much easier to understand if you first thing of simplicial complexes rather than simplicial
sets. Let’s first state the theorem.
Lemma (The Extension Lemma). Let ω be a form in A PL (∂∆n ) then there is a form ω 0 ∈ A PL (∆n ) such that ω 0
maps to ω under the restriction A PL (∆n ) → A PL (∂∆n ).
We’ll prove this in a somewhat gentle way.
Definition. Let X : ∆op → C be a simplicial object in C. We say X is extendable if given a collection of
Φi ∈ Xn−1 such that
di Φ j = d j Φi
i<j
then there exists Φ ∈ An such that Φi = di Φi
Remark. If there is a free-forgetful adjunction F : Set C : U what this is saying is that there is a lift
/ UX
<
∂∆n
∆n
That is, the underlying simplicial set UX is trivially fibrant.
Remark. Yet another way of saying this is that
X (∆n ) → X (∂∆n )
is surjective. In general, this will imply that for any cofibration K ,→ L of simplicial sets, X ( L) → X (K ) is
surjective.
We see from this last formulation that the extension lemma demands that A PL (∆n ) → A PL (∂∆n ). This
will be proved if we know that ( A PL )•• is extendedable. This is what we’re going to prove.
p
Lemma. ( A PL )• is extendedable.
Ok. Now we move to using geometric intuition. There will not really be a way to understand the proof
if you don’t think geometrically.
Let start with a geometric simpliex ∂∆n and a form on it. Lets focus on one face of it. That face has a
form on it. Let’s try to extend it to the whole thing.
[[[[[PICTURE]]]]]
How are these related? Consider stereographic projection from the final vertex:
t0
t n −1
π ( t0 , . . . , t n ) =
,...,
.
1 − tn
1 − tn
(note that stereographic projection onto the rth vertex omits the rth coordinate).
and then consider π ∗ ω |∆n−1 , the pullback of a form
on one of the faces. This form is some giant amalgm
of t0 , . . . , tn−1 , (1−1t
n)
and dt0 , . . . , dtn−1 and d 1−1tn . Note that
1
1
d
=−
dtn .
1 − tn
(1 − t n )2
This means in fact we can eliminate
1
1− t n .
e and N such that
That is, we can choose ω
e
(1 − tn ) N π ∗ ω = ω.
26
e will agree with the form on the lower face, but is a form on ∆n . Now, ω − ω
e |∂∆ will be zero on
This ω
e )|∂∆n and by picking another face.
the face we chose. Now, do this whole thing again starting with (ω − ω
We’ll get a form ω1 on ∆n such that ω − (ω + ω1 )∂∆n is zero on two faces. Keep going.
Now, we won’t be working with topological spaces, so we have to do this in simplicial sets.
Proof of Extension. Suppose we have ω ∈ ( A PL )•• (∂∆n ). We want to extend this to an element of ( A PL )•• (∆n ) =
( A PL )n . We proceed as in the geometric case, but we proceed formally.
We assume we have already extended the first r faces, and we want to extend to the r + 1st.
The argument above involve having fractions, so here we have to invoke fraction fields. We consider
the subalgebra of that fraction field given by adjoining an inverse 1−1 tr .
B0 = h( A PL )0n ,
and we define
d
1
i ⊂ FF (( A PL )0n )
1 − tr
1
1 − tr
=
dtr
(1 − tr )2
to get a CDGA:
B0 ⊗Q Λ(dt1 , . . . , dtn ).
How did things work in the geometric case? We used stereographic projection. In this formal case, pulling
back by stereographic projection amounts to a homomorphism
(
ti
i<r
∗
∗
tr
SP : ( A PL )n−1 → B
SP (ti ) = 1t−
i +1
i ≥ r+1
1− tr
and similarly, restricting to a face is given by the differential dr (1/(1 − tr )) = 1.
Having formally set up the same deal as above, let’s consider SP∗ ωr . We know there is some N such
that
e r ∈ ( A PL )n
(1 − tr )n SP∗ ωr = ω
e r and continue on.
We let ωr+1 = ω
Ok, so at least we have the extension lemma. We’d like to start connecting this picture up with the
picture for chains on spaces. That is, we want a quasi-isomorphism
APL (K ) → C ∗ (K )
and ideally it would also induce equivalences on cohomology rings. We’ll actually give two proofs of this.
This will be a little more algebraic, and then we’ll return to more geometric matters.
We need some intermediate objects. We need chains and piecewise linear chains (which turn out to be
the exact same thing)
Definition. Let K ∈ Set∆ . The rational cochains of a simplicial set are defined as follows. We define
C p (K ) = MapSet (K p , Q)/degenerate simplices
(note these are SET maps). Given f ∈ C p (K )
p +1
(d f )(σ) =
∑ (−1) p+i+1 f (di σ)
σ ∈ K p +1 .
i =1
This has a (non-commutative, but E∞ ) multiplication given as follows
( f ^ g)(σ) = (−1) p f (d p+1 · · · d p+q σ) · g(s0 · · · s0 σ)
Definition. Let K ∈ Set∆ . We define
σ ∈ K p+q
(CPL )•n = C ∗ (∆n )
with obvious face and degeneracymaps. Then define CPL (K ) by our definition above, i.e.
p
p
CPL (K ) = MapSet∆ (K, CPL )
In fact, these objects are really the same thing.
27
Proposition. There is an isomorphism
∼
=
→ C∗ (K )
CPL (K ) −
Proof. This is a kind of shell game. First, we note that we only have to prove it for K = ∆n (why?). So, we
want to produce an isomorphism
p
CPL (∆n ) → C p (∆n ).
But, just using the definitions
p
p
p
CPL (∆n ) = MapSet (∆n , CPL ) = (CPL )n = C p (∆n )
and we’re done.
We now start trying to produce some other isomorphisms. We’ll do it as groups first (which is geometric) and then as rings (that is we’ll show an equivalence once we descend to cohomology rings).
It is easy enough to produce an integration map.
Definition. We first produce an integration map
Z
∆n
: ( A PL )nn → Q
by defining
k1 ! · · · k n !
(k1 + · · · + k n + n)!
(this is what you would get if yiou did this geometrically via a horrible volume integral NOTE: Maybe I
should actually do this)
Z
k
∆n
t11 · · · tknn dt1 · · · dtn =
We also need a definition of pullback:
Definition. Let σ be a k-simplex of ∆n , i.e. σ ∈ (∆n )k , i.e. σ : ∆k → ∆n . Then there is a pullback
σ : ( A PL )n → ( A PL )k
∆k
∆n
given as follows. The map σ :
→
is, by Yoneda, determined by a map σ∆ : [k ] → [n] in ∆. If σ∆ (i ) = j
then σ∗ (t j ) = ti , otherwise σ∗ (t j ) = 0.
Remark. IMPORTANT: We are here using the definition that
( A PL )n = Λ(t0 , . . . , tn , dt0 , . . . , dtn )/
∑ ti = 1, ∑ dti = 1
But when working with differential forms we are kind of assuming that the polynomials are in the variables t1 , . . . , tn . This will get you into trouble if you don’t think carefully about it, because the pullback is
really not operating on t1 , . . . , tn , but on t0 , . . . , tn with relations.
This allows us to make the following definition:
Definition. We define
Z
∆n
ω (σ ) = (−1)
k ( k −1)
2
Z
∆k
σ∗ ω
where ω ∈ ( A PL )kn .
Theorem. The map
R
: ( A PL )•• → (CPL )•• is a quasi-isomorphism.
Proof. We have to show that it’s a chain
R map and also compatible with face and degeneracies. Once this is
done, we’ll be done since we know 1 = 1 and H ∗ (( A PL )n ) = H ∗ ((CPL )n ) = Q.
Let ω ∈ ( A PL )nn−1 . To commute with differentials we must have
Z
Z
dω (σ) = d
ω (σ)
∆n
∆ n −1
Z
=
ω (∂σ )
∆ n −1
n
=
∑ (−1)
i =0
28
n + i +1
Z
∆ n −1
si∗ ω
We may as well assume
k
cj · · · dtn .
ω = t11 · · · tknn dt1 ∧ · · · dt
The left side of this thing may be easily computed
k
cj · · · dtn ) = tk1 · · · k j tk j −1 · · · tkn dt1 ∧ · · · ∧ dtn
d(t11 · · · tknn dt1 ∧ · · · dt
n
1
j
We have have a formula for integrating this — the answer is
kj
k 1 ! · · · ( k j − 1) ! · · · k n !
k1 ! · · · k n !
=
(k1 + · · · + k j − 1 + · · · k n + n)!
( k 1 + · · · + k n + n − 1)
We now compute the right hand side. It’s not as easy. We recall how si∗ works. The map si : [n − 1] → [n]
misses i and si∗ ω we defined above. We note that when ω has the form above si∗ ω = 0 unless i = 0. In this
case
k
[
cj · · · ∧ dtn ) = tk1 · · · tkn dt0 ∧ · · · dt
s0∗ (t11 · · · tknn dt1 ∧ · · · dt
j−1 · · · ∧ dtn−1
0
n −1
At this point we might as well assume j = 0 so that we deal with
k
t01 · · · tknn−1 dt1 ∧ · · · ∧ dtn−1
The problem then becomes eliminating the variable t0 via the relation
t 0 = 1 − t 1 − · · · − t n −1 .
Then do a horrible computation involving multinomial coefficients.
Now we use an argument from Bousfield-Guggenheim to finish this off. It’s really the method of acyclic
models. We begin by having a digression acyclic models.
10.3. The Method of Acyclic Models. This is going to be useful in the proof of the main theorem for
us, and is a general useful thing in algebraic topology. Before we get started, I want to point out some
idiosyncracies in the literature that took me a little while to sort out. Older literature, such as Bousfield
and Guggenheim [], Eilenberg and MacLane [] and Forrester [] have a definition of “representable” functor
which is maybe not very familiar.
Let’s talk about what representable functors actually are, and then we’ll see what they are talking about
in this older literature. I’ve been sloppy and said that a representable functor “is” a functor of the form
HomC (−, c). Here is a better definition.
Definition. A functor F : Cop → Set is representable if there is a natural isomorphism
Φ : HomC (−, c) → F
for some c ∈ C. Here “natural isomoprhism” means that there is another Ψ : F → HomC (−, c) such that
ΨΦ ' ΦΨ ' 1.
Recall that the Yoneda lemma says that such natural transformations (any natural transformation, not
just natural isomorphisms) are in one-to-one correspondence with elements in F (c):
{HomC (−, c) =⇒ F } ↔ F (c)
We can quickly see how this correspondence works.
• Given a natural transofrmation Φ, how do we get an element u ∈ F (c)? Well, just define
u = Φ(IdC )
• Given a u ∈ F (c) and an f : d → c we can also define
Φ( f ) = ( F f )(u).
29
Now, this works for functors into sets. How about chain complexes or some other additive cateogry? If
it was to be literally representable, we would hit the target Set with a free functor:
F
Cop → Set −
→ Ch R .
This is often not the case. Instead, it is more typically that the functor F is a composite of some probe.
That is, there are elements c1 , c2 , · · · ∈ C and we can consider a set F ( f i )ci where c → ci .
Example. Suppose we are given C∗ (−) : Top → ChQ . This is not a representable functor. However, how
do we building Cn ( X )? We look at maps HomTop (∆n , X ) and look at things that are free on this. Of
course, we have to do this in every degree, which is why things aren’t quite representable. But it’s “close
enough” to being representable.
Remark. Such a functor should be thought of as “almost” representable.
This suggests the following definition.
Definition. A functor F : C → Ch R is free with models if there is a set of object M called models such
that for every c ∈ C there is a subset M0 ⊂ M equipped with elements there are em ∈ F (m) for every
m ∈ M0 such that F (c) is free and
M
R h F ( f ) em i
F (c) =
f ∈HomC (m,c)
Now, this definition existed in older literature, but they call it being “representable” and they use all
sorts of weird notation for it. The weird notation comes in because they insist on remembering the natural
isomorphisms that should be around (which is actually good practice). Here’s a better definition, which
maybe you would see in an older book
Definition. Let F : C → Ch R be a functor. Given M as above, define a new functor Fb by
M
Fb(c) =
R h F ( f ) em i
f ∈HomC (m,c)
Then F is representable if there is a natural isomorphism
b
Φ : F → F.
There is a dual definition:
Definition. F : Cop → Ch R a functor. Define
Fb(c) =
∏
R h F ( f ) em i
f ∈HomC (m,c)
It’s basically the same definition. Anyhow, this is important in the acyclic model argument.
I need a few mmore definitions before stating the acyclic model argument. Some will be review. Others
will probably be new.
Definition. Suppose we have two functors F, G : C → Ch R and two natural transformations α, β : F =⇒ G.
Then α, β are naturally chain homotopic if there are chain maps hc : F (c) → G (c) and hc is natural in c,
i.e. the following commutes for f : c → c0
F (c)
hc
/ G (c)
/ G (c0 )
G (c0 )
Definition. We say F : C → Ch R is acyclic on models if H i ( F (m)) is 0 except possibly at i = 0.
Theorem. Let C be a category with models M and suppose F, G : C → Ch R are functors which are 0 in negative
degrees. Suppose also that
• F is free with models M
30
• G is acyclic on M
• l : H0 F → H0 G is a natural transformation of Mod R .
then there is a natural transformation L : F → G inducing l on H0 and it is unique up to natural chain homotopy.
Proof.
Remark. It is natural to break up the proof below into 3 steps
(1) Lifting on models
(2) Lifting on arbitrary objects
(3) Natural chain homotopy
We start the induction by noticing that F0 (c) and G0 (c) are exactly the cycles (since d = 0 because there
is nothing in negative degrees). We get a diagram
F0 (c)
/ H0 F (c)
G0 (c)
/ H0 G (c)
`
Now, it’s easy to produce a map since G0 (c) → H0 G (c) is surjective.
We suppose we are now extending for M. We have a diagram
Fn m
/ Fn−1 m
/ Fn−2 m
Gn m
/ Gn−1 m
/ Gn−2
n ∈ F m. Then we consider the following picture
and we want to lift the left thing. Pick em
n
n
n
/ dem
em
_
n) / L n −1 ( e m
x
/0
n around, and then find that it maps to 0 in G
We just chane em
n−2 ( m ). By the acycliclity (ie. exactness), we
n ) comes from an element in G ( m ). Call this element x and define
then have that Ln−1 (em
n
n
Ln ( em
)=x
We would like to extend the diagram with m replaced by c. But this is now pretty easy since
M
Fn (c) =
n
Rh Fn ( f )em
i
f ∈HomC (m,c)
and we can define this as mapping to
M
n
Rh( Gn f ) Ln (em
)i ∈ Gn (c)
f ∈HomC (c,m)
We can check easily that this commutes with boundaries.
What is left to check is that if we produe two lifts L1 and L2 that they are the same up to natural chain
n,
homotopy. We define the chain homotopy inducively (obviously) by the same game as before. Given em
n
we want to know hem . We don’t know how to do this right now, but we do have the desideratum that
n
n
n
n
) − L2 ( e m
).
dh(em
) + hd(em
) = L1 ( e m
But it’s also easy to check that
n
n
n
hd(em
) − L1 ( e m
) − L2 ( e m
)
is a cycle, so that by acyclicity it is also a boundary. Suppose it is the boundary of some element c. Define
n
h ( em
) = c.
31
Then extend by freeness again.
Corollary. Let F, G : C → Ch R be free and acyclic on M and there is a natural isomoprhis H0 ( F ) ∼
= H0 ( G ). Then
there is a natural isomoprhism H∗ ( F ) ∼
= H∗ ( G ).
10.4. Application: Alexander-Whitney and Eilenberg-Zilber Maps.
10.5. Continuing. We check two things.
Theorem. We have
(1) The rational cochains C ∗ (K; Q) are free and acyclic on models.
(2) The rational differentials forms A∗PL (K ) are free and acyclic on models.
Proof. In this case, the model is ∆n ∈ Set∆ . We know that both are acyclic on models. We need that they
are free on models. The case of cochains is easy, althought maybe we should do it. Let X ∈ Set∆ . We
construct natural transformations
∏
n
C∗ (X ) →
C ∗ ( ∆ n ) m x → C ∗ ( X ).
m x :∆ → X,n
The first natural transformation, Φ is given by
∏
Φ(u) = (C ∗ ( xe)u) x ∈
C ∗ (∆n ) x
u ∈ C∗ (X )
x ∈ Xn
∆n
where x ∈ Xn and xe :
→ X determines the map.
To determine the inverse natural transformation Ψ we need to demonstrate the large product as dual
to C∗ ( X ). We define
*
+
Ψ
∏
mx , σ
= hmσ , ∆n i
x:∆ p → X
We would now like some natural transformations
∏
n
A∗PL ( X ) →
A∗PL (∆n ) x → A∗PL ( X )
m x :∆ → X,n
The first natural transformation, as above is given by
Φ(u) =
∏
n
x:∆ → X
A∗PL ( xe)u
u ∈ A∗PL ( X )
Now we need in an inverse transformation. We need to define Ψ X and it’s enough to define it on
each term. For each m x ∈ ∏ x:∆n p→ X A∗PL (∆ p ) we want to define an element of AnPL ( X ), i.e. a map
φ ∈ MapSet∆ ( X, AnPL ).
If x ∈ Xn , then define φx = m x . Now, if we are working with dimension p and assuming we’ve defined
this map on simplices of lower dimensions. We consider
wi = φ ( d i x ) − d i m x .
This is “extendable” by definition, and suppose it extends to w. We then define φx = w. For degenerate
simplices, we just lift up the definition from lower dimensions. This defines Φ X .
We have to show that ΨΦ = Id.
Let φ0 ∈ AnPL ( X ), i.e. a map X → AnPL and Φ X φ0 = ∏ m x . Then m x = φ0 ( x ). We now work through the
above and note
wi = φ(di x ) − di (m x ) = φ(di x ) − di (φ0 ( x ))
However, we start with φ = φ0 by definition and then we keep going with this.
R
Theorem. The map : A PL ( X ) → C ∗ ( X ) is an isomorphism on homology.
Proof. It’s a natural transformation. These guys are acyclic and free on models.
Actually, we can prove something a little stronger which will be useful.
32
Theorem. Let C•• and D•• be CDGAQ,∆ . Suppose these are quasi-isomorphic and extendable. Then for any K
H ∗ (C (K )) ∼
= H ∗ ( D (K )).
Proof. Acyclic models.
10.6. Multiplicative Matters. This is one more the most interesting parts of rational homotopy theory, at
least to me. This prefigured a lot of math that people are hot and bothered about now (a buzzword would
be “homotopy coherence”). I’m not sure how much I’ll be able to do with this. There’s a lot of machinery
to develop, and even though it is quite nice, I suppose I should move on to proving actual theorems.
So, what I’ll do first is prove an easy version of the theorem we want, and then I might, if I have time,
prove more interesting versions.
Theorem. The rings H ∗ (C ∗ (K )) and H ∗ ( A PL ( X )) are isomorphic (as rings).
Proof. It is enough to show that in the zig-zag
CPL → A PL ⊗ CPL ← A PL
both arrows descend to multiplicative isomorphisms on homology. They are definitely multiplicately.
We prove that c PL is extendable. That is, given a collection in C p (∆n−1 ) we want to cobble together an
element of C p (∆n ). What are elements of C p (∆n−1 )? They are elements of Map(Hom∆ ([ p], [n − 1]), Q).
It’s easy to see how to extend this (quite trivially) to Map(Hom∆ ([ p], [n]), Q).
To see that CPL ⊗ A PL is extendable, we just have to note that each are separately...
11. Geometric Constructions
Our correspondence between spaces and CDGAQ is set up as
X 7→ A PL (Sing X ).
What we’re going to do now is discuss how geometric constructions on the left correspond to algebraic
constructions on the right. We will need to know this information to set up adjunctions that prove an
actual correspondence later on.
11.1. A Quick Example. We now know that H ∗ ( X ) ∼
= H ∗ ( A PL ( X )). This is enough information to actually construct models of spaces.
Example. We’ll consider Sn . We have H ∗ (Sn ) ∼
= H ∗ ( A PL (Sn )), so that there is a generator in degree k,
e in A PL (Sn ).
which we’ll call ω and it has a representative ω
n odd
It is now easy to see that there is a map
f : (Λ(e), 0) → A PL (Sn )
deg e = n
e This map is a quasi-isomorphism.
where we define f (e) = ω.
n even Again, one can easily define a map
(Λ(e), 0) → A PL (Sn )
deg e = n
e But the problem here is that this is not a quasi-isomorphism because Λ(e) is a polynomial
and f (e) = ω.
e 2 is 0 in cohomology, there is a an element α
algebra. We need to kill off e2 . We do this as follows. Since ω
2
0
e = dα. Adjoin an element e of degree 2n − 1 to Λ(e) such that
such that ω
de0 = e2
f (e0 ) = α
This then produces a quasi-isomorphism. This may end up looking quite a bit like π4k−1 (S2k ) to you. It
should.
Example. Let I be the interval. Λ(t, dt) is a Sullivan model for this.
33
Example. Let Y be a space and consider the cone CY = Y × I/I × {0}. There is a chain of quasiisomorphisms
∼/
A PL (CY, ∗)
A PL (Y × I, Y × {0}) o
∼
A PL (Y ) ⊗ A PL ( I, {0})
A PL (Y ) ⊗ Λ+ (t, dt)
And also note
A PL (CY ) ' Q ⊕ A PL (CY, ∗).
Altogether, this gives
A PL (CY ) ' Q ⊕ ( A PL (Y ) ⊗ Λ+ (t, dt))
Remark. I have not said it before, and I should put this a little ealier, but there is a relative construction
A PL ( X, Y ) that fits into a sequence
0 → A PL ( X, Y ) → A PL ( X ) → A PL (Y ) → 0
Example. D n = CSn and we know Sullivan models ΛV for Sn . So, for example, when n is odd the Sullivan
model is Λ(e), deg e = n. And this implies a Sullivan model for D n is
Q ⊕ (Λ+ (e) ⊗ Λ+ (t, dt))
11.2. Pushouts. First, we recall a definition.
Definition. Let A, B, C ∈ CDGAQ with maps A → B and C → B. There is a pullback A ⊗ B C. As a set it
is defined as
A × B C = {( a, c) : f ( a) = g(c)}
Remark. By how we defined A PL (−, −) if we have an inclusino Y → Z and a map Y → X we get a map
A PL ( X ) × APL (Y ) A PL ( Z ) → A PL ( X )
and the fiber of this map is A PL ( Z, Y ) which gives us an exact sequence
0 → A PL ( Z, Y ) → A PL ( X ) × APL (Y ) A PL ( Z ) → A PL ( X ) → 0
Proposition. Let i : Y → Z be an inclusion of a topological space and f : Y → X a map. Then APL ( X ) ×APL (Y )
APL ( Z ) is a model for APL ( X qY Z ).
Proof. We consider the diagram
0
/ A PL ( X q f Z, X )
/ A PL ( X q f Z )
/ A PL ( X )
/0
0
/ A PL ( Z, Y )
/ A PL ( X ) ×
A PL (Y ) A PL ( Z )
/ A PL ( X )
/0
The left downward arrow is a quasi-isomorphism since H ∗ ( Z, Y; Q) ∼
= H ∗ ( X q f Z, X, Q). Take cohomology of everythign and we get long exact sequences. The 5 lemma gives our result.
By properties of homotopy pullback, if we have models ΛV → A PL ( X ), ΛU → A PL (Y ) and ΛW →
A PL ( Z ) then the pullback above is the same as the pullback ΛV ×ΛU ΛW.
There is a special case of this that should be mentioned. That is, when we are pushing out along a cone.
34
11.3. Homotopy Groups. I need to go back and discuss somethign I could have had earlier, but that I
neglected to include (well, not so much neglected as decided we didn’t need it at the time).
Definition. The linear part of a map ϕ : ΛV → ΛW is the map
Q( ϕ) : V → W
is the map defined so that
ϕv − Qϕv ∈ Λ≥2 W
Example. It’s a lame definition, but quite a concrete thing. On generators, morphisms ϕ will be given by
ϕ ( v ) = w + w1 w2 + · · ·
and you just truncate to get the linear part.
Theorem. If ϕ0 is homotopic to ϕ1 as maps ΛV → ΛW between minimal sullivan algebras and if H 1 (ΛV ) = 0
then
Qϕ0 = Qϕ1
Proof. Deferred.
What this says, of course, is that the linear part only depends on homotopy class.
Suppose f : Y → X is a map of simply connected topological spaces. Given models for X and Y we
have a diagram
/ A PL (Y )
A PL ( X )
O
O
/ ΛW
ΛV
We’ll now use this and our model of spheres to define a certain pairing:
Definition. Consider [α] ∈ πk ( X ) given by a representative α : (Sk , ∗) → ( X, ∗). On homotopy, this gives
Q( a) : V k → Q · e.
We then define a pairig
h−; −i : V × π∗ ( X ) → Q
by
(
hv; [α]ie =
Q(α)v
0
v ∈ Vk
deg v 6= deg α
Lemma. This is a bilinear pairing.
Proof. We just need to show linearity in π∗ ( X ). To do this, we need to recall how addition in homotopy
groups is defined. Let [α0 ], [α1 ] ∈ πk ( X ). Then [α0 ] + [α1 ] is defined to be represented by
(α0 ,α1 )
Sk → Sk ∨ Sk −−−−→ X
where the first map is the pinch map. To compute the pairing we need the Sullivan model for A PL (Sk ∨ Sk ).
Of course, H ∗ (Sk ∨ Sk ; Q) has two generators in dimension k, e0 , e1 and may have higher dimensional
generators depending on whether e is even or odd. So, we have a Sullivan model
Λ(e0 , e1 , . . . ) → A PL (Sk ∨ Sk ).
We compute Q for the maps above. The map Sk → Sk ∨ Sk induces a map on Sullivan algebras
Λ ( e0 , e1 , . . . ) → Λ ( e ) .
It is easy to see Q( j)e0 = Q( j)e1 = e.
We also need to comptue Q(α0 , α1 ). It is relatively easy to see that
Q(α0 , α1 )(v) = hv, α0 ie0 + hv, α1 ie1 .
35
Composing everything we get
Q( a)v = (hv, [α0 ]i + hv, [α1 ]i)e
Remark. There is an important consequence of this. The adjoint of the pairing gives
νX : V → HomZ (π∗ ( X ), Q)
Remark. Another quick consequence (which will actually be useful). There is a quasi-isomorphism
Λ ( E ) → ( H ∗ ( S n ), 0)
for n even or odd. In general, we only know the linear part of the map ΛV → ΛE induced by Sn → X.
However, since all non-linear terms are killed off an passing to homology, we in fact know the compositions
ΛV → ΛE → H ∗ (Sn )
since they are given by
v 7→ hv, αi[e]
and it’s multipicative since H ∗ (Sn ) has such a simple algebra structure. This will be used below.
Example. Let’s try CPn . We know that
H ∗ (CPn ; Q) ∼
= Q [ u ] / ( u n +1 )
deg u = 2.
As a first step, we know that since A PL (CPn ) → C ∗ (CPn ) is a quasi-isomorphism that there is some
generator ũ in degree 2. So we would guess at a map
Λ(u) → A PL (CPn )
u2 7→ ũ
But this is not quite right. We also need that ũn+1 should be a boundary since it is 0 in cohomology. Thus,
we need to add another generator, x in degree 2n + 1 such that dx = un+1 :
Λ(u, x ) → A PL (CPn ).
n.
This is a quasi-isomorphism, and so Λ(u, x ) is a model for CPQ
What we’ll see in the future is that Λ(u, x ) contains all information about rational homotopy groups of
CPn .
Example. How about Sk ∨ Sl . We did this a little bit above, but we’ll actually need it up to degree k + l − 1.
The projections maps
Sk ∨ Sl → Sk
Sk ∨ Sl → Sl
give rise to maps
A PL (Sk ) → A PL (Sk ∨ Sl )
A PL (Sl ) → A PL (Sk ∨ Sl )
and we know that
A PL (Sk ∨ Sl ) ' H ∗ (Sk ) ⊕ H ∗ (Sl ) = Λ(e0 , . . . ) ⊕ Λ(e1 , . . . ).
There is an obvious map
Λ(e0 , . . . ) ⊗ Λ(e1 , . . . ) → A PL (Sk ∨ Sn )
and we need to start killing elements. In particular, e0 e1 must be a boundary, so there is an x such that
dx = e0 e1 . All other elements are in higher degree. So,
'
Λ(e0 , . . . ) ⊗ Λ(e1 , . . . ) ⊗ Λ( x, . . . ) −
→ A PL (Sk ∨ Sn )
36
11.4. Cell Attachment. Let X be a simply connected topological space. If we have an element of πn ( X )
we can attach a cell along this map. We’d like to know what this looks like in rational homotopy theory.
Let ΛW be a model for Sn and ΛV a model for X. We already know from our discussion of attaching
cones that that a model for X ∪α Sn is
ΛV ×ΛW (Q ⊕ (Λ+ W ⊗ Λ+ (t, dt)))
Theorem. Let ΛV be a model for X. Define (ΛV ⊕ Qu, dα ) by requiring deg u = n + 1, u · Λ+ V = 0 = u2 and
dα u = 0, dα v = dv + hv, αiu.
This is a model for X ∪α D n+1 .
Proof. We have an quasi-isomorphism
ΛV ×ΛW (Q ⊕ (Λ+ W ⊗ Λ+ (t, dt)))
ΛV × H ∗ (Sn ) (Q ⊕ ( H + (Sn ) ⊗ Λ+ (t, dt)))
We want to produce a quasi-isomorphism from (ΛV ⊕ Qu, dα ) to this.
First, we have a composite
f : ΛV → ΛW → H ∗ (Sn ).
It is enough to define a map from ΛV into the pullback and from Qhui into the pullback.
11.5. The Whitehead Product. We give a definition from good old-fashioned classical homotopy theory.
Definition. Consider Sk × Sl . This is obtained from Sk ∨ Sl by attaching a D k+l . The boundary of this
D k+l is Sk+l −1 . This determines a map
S k + l −1 → S k ∨ S l .
This is the universal (k, l )-whitehead product.
Definition. Let α ∈ πk ( X ) and β ∈ πl ( X ). We define [α, β] ∈ πk+l −1 ( X ) via
(α,β)
Sk+l −1 → Sk ∨ Sl −−→ X.
This gives a product
[−, −] : πk ( X ) × πl ( X ) → πk+l −1 ( X )
We want to look at this from the perspective of rational homotopy theory. What this will end up
corresponding to is some “quadratic part” of ΛV in the same way homotopy groups corresponded to a
linear part of the map.
Definition. The differential d : V → ΛV decomposes into terms such that di (v) is in Λi+1 (V ). It is not
in general true that the di are then differentials (i.e. d2 = 0) although they are still derivations. However,
something special happens for d1 . We have that
d2 − d21 = d1 d2 + d2 d1 + d22 + · · ·
so that d2 − d21 raises word length by at least 3. Thus, −d21 raises word length by at least 3. But this is a
severe problem since d21 should only raise word length by 2. Thus, it must be that d21 = 0 and so d21 is a
differential. It is then the case that (ΛV, d1 ) is a minimal sullivan algebra.
d1 is called the quadratic part of d.
37
12. Main Proof
We begin the proof of the main theorem. This will take the following form. We let Ho(Set)Q−SC be the
full subcategory of the homotopy category on the objects which are rational and simply-connected. We
let Ho(CDGAQ,∗ )min be the full subcategory be given by the minimal algebras of finite Q-type. The main
theorem has the form
Theorem. There is a functor h−i : CDGAQ,∗ → Set∆ such that there is an adjoint
A PL : Set∆ CDGAQ,∗ : h−i
which descends to an adjoint on homotopy categories:
A PL : Ho(Set∆ ) Ho(CDGAQ,∗ ) : h−i
Thus functor descends to an adjoint equivalence of categories
A PL : Ho(Set∆ )Q,SC Ho(CDGAQ,∗ )min : h−i.
12.1. Some More on the Structure of CDGAQ . It turns out that maps between CDGAs can be assembled
into a simplicial set, and we’ll eventually show that this makes CDGAQ a simplicial model category in
the sense of Quillen. What this means is that there is actually a space of maps between any two CDGAs
and that this space plays well in a homotopy theoretic sense.
Definition. Let A, B ∈ CDGAQ . We define the mapping space to be the simplicial set F ( A, B) given by
F ( A, B)n = HomCDGAQ ( A, ( A PL )•n ⊗ B).
The face and degeneracy maps are givne by those on A PL .
Remark. This is actually a kind of amazing thing. If we plug in, for example, B = Q this produces a
simplicial set F ( A, Q) from a CDGA, A. This seems to be inverse to our procedure whereby we took
a simplicial set and turned it into an algebra of differential forms. And indeed, it will be! This is our
machine for turning elements of CDGAQ into spaces! We won’t see how it works quite yet, or how it
plays with rational homotopy equivalences, but we’re on our way.
We prove that this is in fact adjoint to A PL .
Lemma. Let A, B ∈ CDGAQ , W ∈ Set∆ . Then there is a map
Φ : HomCDGA ( A, A PL (W ) ⊗ B) → HomSet∆ (W, F ( A, B))
which is bijective if W is finite or B is of finite type.
Proof. We first define the map. Given f : A → A PL (W ) ⊗ B and given w ∈ Wn we need to produce an
element Φ( f )(w) ∈ F ( A, B)n . And element of F ( A, B)n is a map A → ( A PL )•n ⊗ B. So this is easy:
A → A PL (W ) ⊗ B → A PL (∆n ) ⊗ B
and we’re done. We note that if W is a simplex then this is a bijection. We use this fact to bootstrap up.
We note that there is a presentation of any simplicial set in the following form MOVE TO SIMPLICIAL
SETS. There is a coequalizer
//
/W
ä R ∆deg t−1
ä T ∆deg t
where T is the set of non-degenerate simplices of W and R is the set of rations {di t = s j1 · · · s jk u}.
We’re going to apply to this the functor A PL and then tensor with B and then apply HomCDGAQ ( A, − −
−). We note that A PL takes disjoint unions to products and equalizers to coequalizers. Also, (−) ⊗ B
preserves equalizers and arbirary products if B is finite-type. We thus get an equalizer
A PL (W ) ⊗ B → ∏ T A PL (∆deg(t) ) ⊗ B
38
//
∏ R A PL (∆deg(t)−1 ) ⊗ B
and then
Hom( A, A PL (W ) ⊗ B)
/ Hom( A, ∏ T A PL (∆deg(t) ⊗ B))
// ∏ R Hom( A, A PL (∆deg(t)−1 ) ⊗ B)
HomSet∆ (W, F ( A, B))
∏t∈T HomSet (∆, F ( A, B))
// ∏r∈ R HomSet (∆, F ( A, B)
The last two maps are bijections, so the induced map on the left is a bijection.
Remark. By this adjunction a right homotopy A → A PL (∆1 ) ⊗ B = Λ(t, dt) ⊗ B is the same as an element
h ∈ F ( A, B)1 . This makes sense since a homotopy should be a path in a mapping space.
We now show that this mapping space is homotopically well behaved. This might be a little bit unmotivated, but we’ll see eventually what this buys us. The below is an example of Quillen’s SM7 which tells
us how the simplicial structure on categories plays with the underlying homotopy theory.
Proposition. Let i : V → W be a cofibrtion, p : X → Y in CDGAQ a fibration then the map
(i∗ , p∗ ) : F (W, X ) → F (V, X ) × F(V,Y ) F (W, Y )
is a Kan fibration and a weak equivalence if either i or p is.
Remark. Given the two maps, I can never remember how to right down the space on the right. I always
have to think through the following process. Given a diagram with i a cofibration and p a fibration, there
should be a dotted arrow as indicated
/X
V
>
i
/Y
W
p
Let’s consider the space of maps F (W, X ). There are two natural things to do to it. Push it forward along
p and pull it back along i, giving us the map
F (W, X ) → F (V, X ) × F (W, Y )
But then given a map V → X there is only one thing to do: push it forward to V → Y. Similarly given
a map W → Y we can really only pull it back to a map V → Y. This gives us F (V, X ) → F (V, Y ),
F (W, Y ) → F (V, Y ) and altogether
F (W, X ) → F (V, X ) × F(V,Y ) F (W, Y )
Proof. This is an exercise in adjointness. By definition of Kan fibration, we have to show that for every
Λin ,→ ∆n we have the following lifting
Λin
/ F (W, X )
6
∆n
/ F (V, X ) ×
F (V,Y ) F (W, Y ).
This is equivalent to having a lifting in a diagram
Λin
/ F (W, X )
;
/ F (V, X )
4
∆n
/ F (W, Y )
/ F (V, Y )
39
Now we play with adjointness. The above lift is equivalent to a lift in a diagram
V
/ A PL (∆n ) ⊗ X
9
W
/ A PL (∆n ) ⊗ Y
∼
∼
/ A PL (Λin ) ⊗ X
3
/ A PL (K ) ⊗ Y
Now, the indicated maps are homology equivalences and surjections. It’s now easy to get a lift. The
second part is easy (left as an exercise).
We can now reap useful, an more concrete looking corollaries.
Corollary. Let i : V ,→ W be a cofibrations. Then F (W, X ) → F (W, V ) is a Kan fibrations.
Corollary. If W is cofibrant then F (W, X ) or is a Kan complex (i.e. fibrant in Set∆ ).
Proposition. Let W ∈ CDGAQ be cofibrant and f : X → Y a weak equivalence. Then
F (W, f ) = f ∗ : F (W, X ) → F (W, Y )
is a weak equivalence in simplicial sets.
In our study of spaces we’ll deal with homotopy groups. This means, implicitly, that we are working
with pointed spaces. We will thus need a notion of pointed CDGA and a concomitant notion of function
space.
Definition. An element of CDGAQ,∗ is an element of CDGAQ together with an augmentation e : A → Q
Definition. F0 ( X, Y ) := ker( F ( X, Y ) → F ( X, Q))
Remark. We could also write it as
e Y)
F∗ ( X, Y )n = HomCDGAQ,∗ ( X, ( A PL )•n ⊗
where
e Y = (Q ⊗ Q) ⊕ (( A PL )•n ⊗ Y )
( A PL )∗n ⊗
This also suggests how to define a homotopy for augmented algebras
Definition. f : A → B in CDGAQ,∗ are homotopy if there is h ∈ F∗ ( A, B)1 such that d1 h = f and d0 h = g.
e Y in CDGAQ,∗ .
This is equivalent to a map h : A → Λ(t, dt)⊗
Definition. Let S(n)∗ ∈ CDGAQ,∗ be the pointed sphere.
12.2. Homotopy Groups of CDGAs. We need to define homotopy groups of CDGAs. They work like the
homotopy groups of anything really. They detect weak equivalences. It will just be useful to have these,
and it turns out that they’ll have some correspondence between homotopy groups of spaces. In fact, we’ve
seen these things before.
Definition. The n-th homotopy group of A where A ∈ CDGAQ,∗ is given by
π n A = H n ( QA)
Q = A/A · A
and A = ker( A → Q).
Proposition. Let f : A → B be a map of pointed CDGAs. And suppose f ' g. Then
f ∗ = g∗ : π ∗ A → π ∗ B.
40
Proof. We want H n ( QA) → H n ( QB) an isomorphism. We know that there is a homotopy h : A →
e B. There are two maps d0 , d1 : Λ(t, dt) → Q and we look at the composites
Λ(t, dt)⊗
Q ⊗ QB
7
/ Λ(t, dt) ⊗ QB
A
'
Q ⊗ QB
We can look at this on homology and note that (d0 )∗ = (d1 )∗ since these are maps
H ∗ (Λ(t, dt) ⊗ QY ) → H ∗ (Q ⊗ QY )
(this is by Kunneth).
∼
Proposition. Suppose f : A −
→ B with A, B Sullivan. Then π ∗ A → π ∗ B is an isomorphism.
Proof. For maps between fibrant-cofibrant things, weak equivalence implies homotopy equivalence, an
then use the above.
Proposition (Mayer-Vietoris). Suppose we are given a pushout diagram in CDGAQ,∗
V _
W
j
/X
k
h
/Y
where i is a cofibration. then there is a long exact sequence
(i∗ ,j∗ )
π0 V
/ π 0 W ⊕ π 0 Xh∗ −k∗
/ π0Y
/ π1Y
π1 V
s
/ π1W ⊕ π1 X
π2 V
s
/ ···
Proof. The Mayer-Vietoris sequence is usually something that happens with homology, so we try to get to
a point where we can do this with homology. So, it’s enough to show that
QV
/ QX
QW
/ QY
is a pushout of chain complexes with Qi : ( QV )n → ( QW )n is an injection. So we prove this. We let
D (n)∗ ∈ CDGAQ,∗ be the augmentated acyclic complex. That is, it’s the algebra with
(1) D (n)∗ · D (n)∗ = 0
(2) H ∗ ( D (n)∗ ) = 0
(3) D (n)∗ is only non-zero in degree n − 1, n
It is easy to check that there is a bijection
HomCDGAQ,∗ ( X, U (n)) ∼
= HomQ (( QX )n , Q).
41
Why? The only non-zero stuff are maps X n → Q and X n−1 → Q. We choose choose map assigning
elements of X n to elements of Q. Since the map overall must be multiplicative, anything that is decomposable in X n must map to 0. If x ∈ X n is of the form x = dy for y ∈ X n−1 then f y is fully determined by
the requirement that the diagram
Xn o
d
Qo
∼
=
d
X n −1
Q
commute.
We now have
HomCDGAQ,∗ (W, D (n)∗ ) ∼
= HomCDGAQ,∗ (V, D (n)∗ )
= HomQ (( QW )n , Q) → HomQ (( QV )n , Q) ∼
and this is surjective since there is always a lift in the diagram
V
/ D (n)∗
W
/Q
This means that ( QV )n → ( QW )n is injective.
Definition. When we have a pushout diagram as above with X = Q we have
V
/Q
W
/ WV
where we are then forced to define
W V = W/(iV ) · W.
Corollary. There’s a long exact sequence
/ π0W
π0 V
π1 V
s
/ π 0 (W V )
/ ···
12.3. The Adjoints.
12.4. The Black-Box.
13. Quillen’s Model
14. Appendix: Simplicial Sets
Definition. The simplex category ∆ has
• objects: ordered sets [n] = {0 < 1 < · · · < n}
• morphisms: functions that preserve order.
This, however, isn’t a very useful definition for us.
Definition. The simplex category ∆ has morphisms generated by
• coface maps: These are given by maps di : [n − 1] → [n], the injectino that doesn’t hit i.
• codegeneracy maps: These are maps si : [n + 1] → [n] with si (i ) = si (i + 1) = i.
42
These satisfy the relations, called the simplicial identities
d j ◦ d i = d i ◦ d j −1 i < j
s j ◦ s i = s i ◦ s j +1 i


 d i ◦ s j −1
s j ◦ di = Id


d i −1 ◦ s j
≤j
i<j
i = j i = j+1
i > j+1
For simplicial sets, we want to use ∆op . Unfortunately, that “op” creates a bit of an issue in terms of our
feeble human minds thinking about the category. So, here’s a presentation that is useful for remembering
relations
Definition. Let (for lack of a better name) L−,+ be the category with
• objects: Are ordered finite sets with two distinguished poitns − and +. For example, [2] is −1, 2+
• morphisms: Are maps of sets that preserve − and + and otherwise preserves ordering.
Proposition. The categories ∆op and L−,+ are equivalent.
Proof. An equivalence of categories is a functor which is essentially surjective and fully faithful. We’ll do
better than this.
The object [n] corresponds to [n − 1]−,+ .
It suffices to give the correspondence on morphisms between the generators of ∆op .
The coface maps di : [n − 1] → [n], which present as maps [n] → [n − 1] in ∆op correspond to maps
[n]−,+ → [n − 1]−,+ as follows. A map [n − 1] → [n] misses the ith spot. We define [n]−,+ → [n − 1]−,+ as
the unique linearly ordered map where the ith spot is hit twice.
The codegeneracy maps si : [n] → [n + 1] which present as maps [n + 1] → [n] in ∆op correspond to
maps [n + 1]−,+ → [n]−,+ . The codegeneracy maps hit the ith spot twice. In [n + 1]−,+ → [n]−,+ we
simply miss the ith spot.
Example. [1] → [2] degeneracy and [3] → [2] face maps. DRAW.
Definition. A simplicial set is a functor X : ∆op → Set. Maps between simplicial sets are given by natural
transformations. We note the category of such things Set∆
In fact, there are broader definitions:
Definition. Let C be a category. A simplicial object in C is a functor X : ∆op → C. Or it’s a functor
X : SO−,+ → Set.
Example. One can have simplicials spaces, simplicial groups, simplicial rings, simplicial spectra, etc.
Remark. Of course, the definition above is very slick, but it’s not the only way of presenting things.
Frequently, simplicial sets are given by combinatorial data. That is a simplicial set is a collection of sets
Xn together with face and degeneracy maps
d i : X n → X n −1
s i : X n → X n +1
such that
d i ◦ d j = d j −1 ◦ d i
i<j
s i ◦ s j = s j ◦ s i −1


 s j −1 ◦ d i
di ◦ s j = Id


s j ◦ d i −1
i>j
There are important canonical simplicial sets
43
i<j
i = j, i = j + 1
i > j+1
Definition. We define ∆n to be the simplicial set co-represented by [n]:
∆n [k] = Hom∆ ([k ], [n]).
This is the Yoneda embedding
∆ ,→ Set∆
op
Lemma. We have
HomSet∆ (∆n , X) = X[n].
That is, maps from ∆n to X are in correspondence with the n-simplices of X.
Proof. This is the Yoneda lemma.
This is all well and good, but how should one think of a simplicial set? We should really think of them
as geometric objects.
Example. ∆2 . This has one non-degenerated 2-simplex, 3 non-degenerate 1-simplices and 3 non-degenerate
0-simplices. Let’s see this. ∆2 [0] = Hom∆ ([0], [2]). There are obviously 3 such maps, and thus 3 elements
in ∆2 [0]. ∆2 [1] = Hom∆ ([1], [2]). There are 3 (non-degenerate) elements. Similarly ∆2 [2] = Hom∆ ([2], [2])
and there is only 1 (non-degenerate simplex).
Example. Let C be a category. Any old category. We define the n-simplices to be compositions
N (C) n = { c0 → · · · → c n }
The face maps are given by composition and degeneracy is given by inclusion. To be more specific, we
have si : NCn → NCn+1 by repeating ci :
( c0 → · · · → c n ) 7 → ( c0 → · · · → c i → c i → · · · → c n )
We also have di : NCn → NCn−1 which composes arrows
( c 0 → · · · → c n ) 7 → ( c 0 → c i −1 → c i +1 → · · · → c n ).
Example (Example above revisited). We note that n-simplices of N (C) in fact correspond to maps ∆n →
N (C). Of course, ∆n itself corresponds to the nerve of [n] considered as a category N ([n]). Thus, nsimplices of N(C) really correspond to functors [n] → C.
Remark. I want to say here why we even deal with degeneracies. It kind of seems like we don’t need them.
After all, when we are thinking of things in a geometric way it kind of seems like we don’t need them.
But there is an easy argument that one may want them. Suppose we consider a projection π : ∆2 → ∆1 .
How would we do this? By definition of morphisms of simplicial sets, we would need 2-simplices to map
to 2-simplices. But without degeneracies, what would the 2-simplex map to in ∆1 ?
Since I’ve been talking about simplicial sets, perhaps I had best talk a little bit about morphisms. Of
course, since a simplicial set is a functor ∆op → Set∆ it’s easy enough to define a morphism to be a natural
transformation. Of course, that’s not particularly illuminating. It’s better to just see that a simplicial map
X → Y is a map of sets Xn → Yn such that the face and degeneracy maps are preserved. But there’s not
much more to it than that. The key thing is really that the degeneracies have to be in there or we can’t get
things like the projection map, or any map that changes “dimension” really.
Remark (Miscellaneous Remark). This is as good a point as any to point out an important fact. Let
F : Set∆ → C be a colimit preserving functor from simplicial sets to some category C (which obviously has
colimits...). For example, F could arise as a left adjoint. It is a fact that in the category Set∆ we have
X = colim∆n → X ∆n .
Think about this statement and why it is true. But, the nice thing about this is that if we want to prove
something about F ( X ), it often suffices to prove the same thing about F (∆n ) and observe that F ( X ) is a
colimit of F (∆n ).
44
14.1. Geometric Realization and Model Structure. There is a model structure on simplicial sets (and
this will in fact be useful for us later), but before we discuss this we need a discussion of an important
construction with simplicial sets. Simplicial sets, as one can imagine from the above constructions, are
combinatorial ways of repesenting topological spaces. At least, that’s how it’s best to think of them in our
brains. But is there a way in which this is really true? Yes. We can turn a simplicial set into a topological
space. We first discuss a more general construction that is a little easier to keep in ones brain.
There is a reasonable general way of “tensoring” two kinds of objects together.
Definition. Let X : I → C and Y : Iop → C then assuming C has appropriate products and colimits, we
can define
X ⊗I Y :=
ä X (c) × Y (c)/(x, f y) ' (x f , y)
= coeq
ä X (c) × Y (d) ⇒ ä X (c) × Y (c)
!
c
c,d
Remark. I learned about this construction in one of my favorite papers [?].
This is quite a useful construction — it is a generalization of many constructions which we are already
familiar with.
Example. ∗ ⊗I Y = colim Y.
Example. If X : C → Top is a diagram of topological spaces, then
hocolimC X = B(C ↓ Id) ⊗C X
Example. We now define a functor
∆• : ∆ → Top
that we’ll use to define geometric realization. We let
n
∆ = ∑ t i = 1 t0 , . . . , t n ≥ 0 .
What are the (co)face and (co)degeneracy maps? The coface maps di : ∆n−1 → ∆n is given by inserting a
0 in the ith coordinate. The codegeneracy map si : ∆n+1 → ∆n is given by
( x 0 , . . . , x n +1 ) 7 → ( x 0 , . . . , x i + x i +1 , . . . , x n ).
This furnishes us with another example of a simplicial set
Example. Let
Sing( X )n = MapTop (∆n , X ).
This is a simplicial set, since there are face maps
di : MapTop (∆n , X ) → MapTop (∆n−1 , X )
and degeneracy maps
si : MapTop (∆n , X ) → MapTop (∆n+1 , X )
We are now in a position to define geometric realization.
Definition. Given a simplicial set X : ∆op → Set we extend X to a map to topological spaces via the
discrete topology:
e : ∆op → Set ,→ Top.
X
We also have the cosimplicial topological space ∆• : ∆ → Top. The geometric realization is given by
| X | := X ⊗∆ ∆• .
45
Remark. What is this, concretely? It’s a coequalizer

coeq 
ä
[m]→[n]

Xn × ∆ m ⇒ ä Xm × ∆ m 
= ä Xm × ∆ m
m
( x, di p) ' (di x, p), ( x, si p) ∼ (si x, p)
m
where the equivalence relation is given by
The way to think about it is to that you’re assigning every element of Xn to a geometeric n-simplex,
and then gluing according to how things are glued FINISH
We need a few more examples of simplicial sets.
Example. Horns. Let Λin denote the simplicial set that is the union of all faces of ∆n except the ith face.
Example. Simplicial spheres. These are ∂∆n generated by the simplicial set which is the union of all the
(n − 1)-faces.
We are now in a position to state the model structure on simplicial sets. It is somewhat hard to actually
prove this, but easy enough to state
Theorem. Set∆ is a cofibrantly generated model category with
• generating cofibrations
I = {∂∆n ,→ ∆n : n ≥ 1}
• generating acyclic cofibrations
J = {Λin ,→ ∆n : n ≥ 1, i }
• Weak equivalences: X → Y is a weak equivalence of simplicial sets if | X | → |Y | is a weak equivalence of
topological spaces.
We record various consequences of this theorem.
Corollary. Cofibrations of simplicial sets are levelwise injections.
Corollary. Fibrations are exactly the maps X → Y that have the right lefting property with respect to J:
Λin
/X
?
∆n
/Y
These are called Kan fibrations.
Corollary. Acyclic fibrations are exactly the maps X → Y that have the right lifting property with respect to J
∂∆n
/X
=
∆n
/Y
15. Appendix: Model Categories
Before we begin, a good reference for this too brief section is [?].
Model categories are types of categories that allow us to do homotopy theory. They are a model of a
homotopy theory. What sort trappings exist for homotopy theory? We would definitely like to have the
concept of a homotopy, so in a category C, we’d like to have a notion of “cylinder” (for a topological space
X, X × I is the cylinder). We’d also maybe want a notion of path object. And if we further look at what
happens for topological spaces, we’d maybe want fibrations and cofibrations. This lead to the definition
of model category.
46
Definition. A model category is a category C closed under (countable?) limits and colimits, together
with the data of three subcategories, cofibrations co(C), fibrations fib(C), and weak equivalences w(C)
and satisfying the following axioms (we define trivial cofibrations to be co(C) ∩ w(C) and trivial fibrations
fib(C) ∩ w(C))
(1) (2 out of 3). If any two of f , g, g f are weak equivalences, then the third is.
(2) (Retracts) Retracts of cofibrations, fibrations and weak equivalences are cofibrations, fibrations of
weak equivalences, respectively.
(3) (Lifting) Let i be a trivial cofibration and p be a fibration below:
A _
/X
?
i ∼
/Y
B
p
Then the dotted arrow exists. This is usually expressed by saying “trivial cofibrations have the
left-lifting property with respect to fibrations”.
Similarly, if i is a cofibration and p is a trivial fibration, then there is a lift in the following
diagram
/X
A _
?
i
∼ p
/Y
B
This is usually expressed by saying “cofibrations have the left lifting property with respect to trivial
cofibrations.”
(4) (Functorial Factorization) Every morphism f : X → Y can be factored as a
X ,→ Y 0 → Y
where the first map is a cofibration and the second map is a trivial cofibration. Further, every
morphism can be factored as
X ,→ X 0 → Y
where the first map is a trivial cofibration and the second is a fibration.
This is all one needs to do homotopy theory! That fact shouldn’t be obvious yet, but it will become
clearer. As we develop these, one should keep in mind the example of topological spaces. In this case
cofibrations are what we usually call cofibrations, and same for fibrations. Weak equivalences are weak
homotopy equivalences. The fact that topological spaces satisfy all the requirements of a model category
is usually proved (without really saying it) in a standard course on algebraic topology. But note! It’s not
an easy formal thing. And this will usually be the case for model categories — actually producing a model
structure on a category is a little difficult.
Before going on we need a little more verbiage. First, note that model categories automatically have an
initial object 0 and a terminal object ∗.
Definition. An object X is called cofibrant if 0 → X is a cofibration. An object Y is called fibrant if Y → ∗
is a fibration.
Remark. If we want to think about the example of topological spaces, it turns out that cell complexes are
cofibrant and that all topological spaces are fibrant.
Definition. Given a morphism 0 → X we can apply functorial factorization to get a composition
0 ,→ QX → X
where the first map is a cofibration and the second map is a tirival fibration. Thus, we have an object QX
that is fibrant and weakly equivalent to X. This is called a cofibrant replacement.
47
Definition. Given a morphism Y → ∗ we can apply functor factorization to get a composition
Y ,→ RY → ∗
where the first map is a trivial cofibration and the second map is a fibration. Thus, we have an object RY
that is fibrant and weakly equivalent to Y. This is called a fibrant replacement.
We pause here for something rather useful. This gets invoked all the time, so often that it doesn’t get
referenced explicitly.
Lemma (Ken Brown’s Lemma). Let C, D be model categories. Suppose there is a functor F : C → D such that
'
'
'
'
F (cof ,−
→ cof) =−
→
then
F (cof −
→ cof) =−
→
'
Proof. This will just use all the axioms. Suppose we have a weak equivalences A −
→ B between cofibrant
objects. We factor into
p
 q
/ / B.
/C
AqB
Note that both A ,→ A q B and B ,→ A q B are both cofibrations. We have a diagram

*/ C ' 3 / B
/ A q B 
A
'
By the 2/3-axiom, A ,→ C is a trivial cofibration (of cofibrant objects), so by assumption F ( A ,→ C ) is a
weak equivalence. However
F ( B ,→ A q B ,→ C → B) = F (Id)
Thus F ( p : C → B) is a weak equivlance. So,
F ( A ,→ A q B ,→ C → B)
is a weak equivalence.
We’ve claimed that model categories are a place to do homotopy theory. So there had better be a way
to describe what a homotopy theory is. When we’re working with topological spaces, we usually describe
this as a map X × I → Y with certain restrictions. That is, it is map out of a cylinder.
Definition. Let X ∈ C be an object. A cylinder object for X is a factorization of the map X q X → X:
∼
X q X ,→ Cyl( X ) −
→ X.
where the first map is a cofibration, and the second is a weak equivalence.
Note the cylinder object comes with two inclusions i0 , i1 : X → Cyl( X ).
Dually, we have the following:
Definition. Let Y ∈ C be an object. A path object for Y is a factorization of the diagonal map Y → Y × Y:
Y ,→ PY → Y × Y.
where the first map is a weak equivalence and the second map is a fibration.
Note that there are two projections π0 , π1 : PY → Y × Y.
Remark. For topological spaces, this is exactly the usual path object.
Remark. Since factorizations are functorial, we can choose cylinder and path objects functorial. We normally denote the functorial cylinder object by B × I and the path object by X I . Note, however, that these
are just particular examples of cylinder and path objects. Homotopies don’t necessary have to involve these
particular path and cylinder objects. We can use whatever path and cylinder objects we want to define a
homotopy. This isn’t so surprising. We can equally well define the concept of homotopy in topological
spaces with X × [0, 1] as we can with X × [0, 3].
We can now finally define what a homotopy should be.
48
Definition. Let f 0 , f 1 : X → Y be maps in C.
(1) f 0 is left homotopic to f 1 if there is a map H : Cyl( X ) → Y such that Hi0 = f 0 and Hi1 = f 1 . The
map H is called a left homotopy
(2) f 0 is right homotopic to f 1 if there is a map K : X → PY such that p0 K = f 0 and p1 K = f 1 .
Definition. f 0 is homotopic to f 1 , written f 0 ∼ f 1 , if f 0 is both left and right homotopic to f 1 .
We also have a notion of homotopy equivalence
Definition. f : X → Y is a homotopy equivalence if ther eis a map g : Y → X such that g f ∼ IdY and
f g ∼ IdX .
The following will be conceptually useful in our discussion of Sullivan algebras. One may hope that if
you have f , g : A → B with f homotopic to g and h : B → X then h ◦ f is homotopic to h ◦ g. This turns
out to mostly be true, but one has to consider handedness.
Theorem. Let C be a model category and f , g : B → X who maps. If f 'l g and h : X → Y is another map, then
h ◦ f 'l h ◦ g. Similarly, if we have a map e : A → B and f 'r g then f ◦ e 'r g ◦ h.
Proof. We know that f 'l g so that there is a composition B q B → B × I → X with Hi0 = f and Hi1 = g.
Simply post-composing we get
BqB → B×I → X →Y
and this gives the required homotopy.
Similarly, if f 'r g we have a composition
K : B → XI → X × X
so that π0 K = f and π1 K = g. By precompsoing we have
A → B → XI → X × X
that gives the required homotopy.
Of course, it would be very nice if we didn’t have these handedness issues. So, in the first case above,
we’d like to have a map IN, and the resulting composition still be homotopic.
Proposition. Let f , g : B → X be maps in C.
Suppose X is fibrant. Then, if f 'l g and h : A → b is a map f ◦ h 'l g ◦ h.
Suppose B is cofibrant. Then if f 'r g and e : X → Y is a map then e ◦ f 'r e ◦ g.
Proof. We do one case and leave the other. We have a left homotopy, presented as
H : B0 → X
where B0 is a cylinder functor, which means there is a factorization
∼
B q B ,→ B0 −
→B
where B q B → B0 is a cofibration and B0 → X is a weak equivalence. We now factor the map B0 → B:
B0 ,→ B00 B
where the first map is a trivial cofibration and the second map is a fibration. However, by the 2/3 axiom,
it must be that the second map is a weak equivalence. Also, since cofibrations compose, B ,→ B00 is a
cofibration. Thus, the factorization
B q B ,→ B00 B
00
displays B as a cylinder object where the map to B is a trivial fibration. Further, we have a diagram
B0
/
>∗
B00
/X
49
and a lift by the fact that B0 toB00 is a trivial cofibration and X is fibrant. Thus, our homotopy may be
presented as H : B00 → X.
We’d like to now have a map out of a cylinder object of A (so that we get a homotopy). Form the
diagram
fqf
A q _ A
/ BqB
/5 B00
/A
/B
∼
A0
Lemma (Equivalence Relation). Let B be a cofibrant object in C. Then left homotopy is an equivalence relation on
MapC ( B, X ) (and dually for fibrant objects).
Proof. Neeed to prove that the left homotopy is reflexiv, symmetric and transitive. Suppose f '` g : B → X
via a cylinder object B q B ,→ B0 → b. Given a homotopy H : B0 → X to get a reverse homotopy we can
just flip B q B. So, it’s obviously symmetric.
To see transitivity, we will need B cofibrant. Suppose f 'l g and g 'l h as maps B → X. We assume
these are given by homotopies H f ,g : B0 → X and Hg,h : B00 → X. We want to stitch these together
somehow, as we would for homotopies of topological spaces. Form the pushout square in the diagram
below
i0
 i00
/ B00 o 1 B
B
i1
BO 0
/L
i0
B
The face inclusions for the new cylinder are the composites
i0
B−
→ B0 → L
i0
1
B−
→
B0 → L
'
These together give B q B → L −
→ B and we also have a map H 00 : L → X. However! We don’t know that
it is left homotopy, since it is not yet clear that L is a cylinder object. But, factor B q B → L as
'
'
B q B ,→ L0 −
→L−
→B
and this L is a cylinder object.
Lemma. Suppose B is cofibrant, h : X → Y a trivial fibration OR a weak equivalence between fibrant objects. Then
∼
=
MapC ( B, X )/ 'l −
→ MapC ( B, Y )/ 'l
'
Proof. One follows from the other by Ken Brown’s lemma. Suppose X −
→ Y is a trivial fibration. We want
to show the map is surjective. Suppose we have f 0 : B → Y a map. We can lift
?X
'
/Y
B
by the axioms (since B is cofibrant). So much for surjectivity.
50
We now need injectivity. Suppose h f 'l hg for two maps f , g : B → X via a homotopy H : B0 → Y.
Form the diagram
BqB
f qg
/X
<
h
B0
/Y
H
then the lift is a left homotopy (the lift exist since the vertical left map is a cofibration).
Lemma. B cofibrant, then f 'l g implies f 'r g.
Dually, if X is fibrant, f 'r g implies f 'l g.
Proof. Suppose B is cofibrant, f , g : B → X and H : B0 → X a left homotopy. Choose a path object PX for
X. Form the diagram
B _
f
/X
i0
B0
r
J
( f g,H )
6/ PX
/ X×X
Then consider
i1
J
B ,−
→ B0 −
→ X0 .
This is a right homotopy.
Corollary. C a model category, B cofibrant, X fibrant, then left and right homotopies are equivalence relations on
MapC ( B, X ).
Corollary. The category Cc f / ' exists.
Proposition. C a model category. Then a map in Cc f is a weak equivalence if and only if it is a homotopy equivalence.
Proof.
15.1. The Small Object Argument.
Exercise. Go look up ordinals, successor ordinals, and anything involving infinite ordinals. Feel uncomfortable about the foundations of mathematics.
We want to begin to develop the machinery for actually building model categories. We certainly defined
them, and saw that if they existed then they are a reasonable context in which to do homotopy theory. But
we don’t have any examples yet.
The best way to build examples turns out to be via cofibrantly generated model categories. The point
of these categories is that the are distinguished classes of maps which we can get our hands on, and which
in some sense generate all the cofibrations. In order to build a model category, we of course have to make
sure that all of the axioms are satisfied. It turns out that the most interesting one to get is the factorization
axioms — this will require some careful discussions of size in categories.
We need some discussion about transfinite composition and the like. This is not just a ridiculous exercise. We use this all the time in homotopy theory without really thinking about it. For example, suppose
you build some cell-complex by attaching infinitely many cells. Then, suppose after that construction you
glom on infinitely many more cells. You’ve tacitly used the ordinal ω + ω but never really discussed it.
For our purposes, I should note, we consider ordinals as categories. Ordinals are of course well ordered
sets, and we can make such a thing into a category by considering morphisms from smaller to larger
elements of the set.
Remark. In the below, we’ll assume the von Neumann model of ordinals.
Example. picture of ω + ω
51
Definition. Let λ be an ordinal. A λ-sequence in a category C is simply a colimit-preserving functor
X : λ → C.
The map X0 → colimβ<λ X β is called a transfinite composition.
The following definition, though weird and innocuous-seeming, is actually crucial for our arguments
below.
Definition. Let κ be a cardinal. An ordinal λ is κ-filtered if
(1) It is a limit ordinal
(2) For any set S ⊆ λ with cardinality |S| ≤ κ, we have sup(S) < λ.
Let’s unpack what this is saying.
Example. Suppose κ = 10. What does it mean to be 10-filtered? I pick any set S ⊂ λ of size 10. I look
at its supremum. I want sup(S) < λ. How can I guarantee that this will supremem will be less than the
ordinal? Only if the ordinal is a limit ordinal. This is, of course, true for any finite n.
Example. What about when κ = ℵ?
Definition. Let C be a category with all small colimits and M ⊆ Mor(C) a collection of morphisms. Let
A ∈ C and κ be a cardinal. Then A is κ-small relative to M if for any κ-filtered ordinal λ and all X : λ → C
with X β → X β+1 in M
colimβ<λ MapC ( A, X β ) → MapC ( A, colimβ<λ X β )
Remark. What is this saying? More or less that if A is κ-small, then if there is a map A → colimβ<λ X β ,
the map factors through some stage A → X β .
There is further verbiage.
Definition. If we don’t specify the cardinal, but just assume there is some cardinal so that the above is
true, we call A small relative M. If further, M = Mor(C) we just call A small.
Proposition. Every set is small.
Ok, now we get to why we are even considering these things. The point is we want to be able to do
infinite constructions and have some actual handle on what the hell is going on.
The following is a key definition. It will take a little while for it to sink into memory, and perhaps not
until we see examples of model categories.
Definition. Let I be a class of maps in a category C. A map f is I-injective (and the set of such is called
I-inj) if
I R f
A map f is is an I-cofibration if
f L I-inj
Remark. I can only remember this by thinking of a picture
I R I-inj I-cof L I-inj.
I now come to why I’ve been rambling about “cells” for a while. The following concept comes up in
many guises (well, the many guises were really the inspiration, but once we have the concept you’ll see it
everywhere).
Definition. Let I ⊂ Mor(C) and suppose C has all small colimits. A relative I-cell complex is a transfinite
composition of pushouts of along elements of I.
Remark. That is, you form a pushout
A
/X
B
/ X qA B
and you do it a whole bunch of times.
52
A quick consequence.
Lemma. I-cell ⊆ I-cof.
We are now in a position to do the small object argument. It is very clever.
Theorem. Let C be a category with all small colimits and I ⊂ Mor(C). Suppose the domains of I are small relative
to I-cell. Then there is a functorial factorization
( F, G ) : Mor(C) → Mor(C) × Mor(C)
such that F ( f ) ∈ I-cell and G ( f ) ∈ I-inj.
Remark. If the domains of I are small relative I-cell it is often said that I permits the small object
argument.
Proof. The domains are small relative I. Assume they are κ-small and let λ be any κ-filtered ordinals. The
goal is to define a λ-sequience Z λ−seq : λ → C so that Z = colim Z f factors the map.
We start off with Z0 = X. Suppose we’ve defined Zα for all α < β where β is a limit ordinal. Then
define
Zβ = colimα< β Zα .
Suppose we want to define something for a succesor ordinal. That is, we’ve defined Zβ and want Zβ+1 .
HERE IS THE CORE OF THE PROOF. Let S be the set of commutative diagrams
A
/ Zβ
B
/Y
then for every diagram of this form, define Zβ+1 by a pushout
äs∈S As
/ Zβ
äs∈S Bs
/ Z β +1
Then, form Z = colimβ<λ Zβ . The induced map Z → Y will have the RLP w/r/t I. How do we see
this? Look at big square in
/ Zβ
A
Z
?
∈I
/Y
B
Since the domains of I (e.g. A) are κ-small relative I-cell, A → Z factors through some Zβ (as in the
diagram). Now, the diagram involving A, B, Zβ , Y is exactly one of the sort that we were dealing with
before. So there is a lift as indicated. This provides a lift and then Z → Y is in I-inj.
This is somewhat important, as it will later give a characterization of cofibrations.
Corollary. If I permits the small object argument, then if f : A → B is in I-cof, there is a g : A → C in I-cell such
that f is a retract of the following form:
A
g
B
A
/B
53
A
f
g
/B
Proof. Factor A → B as
I-inj
I-cell
A −−→ C −−→ B.
Since f is in I-cof it has the LLP w/r/t to the map C → B:
A
/C
?
B
B
I-inj
If you rearrange this diagram, you get the one in the statement of the corollary.
Remark. The sort of argument above is quite standard and is called the retract argument.
15.2. Cofibrantly Generated Model Categories.
Definition. M will be a cofibrantly generated model category if there are sets I, J such that
• The domains of I, J are small relative I-cell and J-cell respecitively.
• fib(M) = J-inj
• w ∩ cof(M) = I-inj.
Remark. It will be convenient to denote such a structure as a quadruple (C, w(C), I, J ).
Of course, I’ve just said what this is. I haven’t shown they exist or how to construct them or we why
like them. Before I do that, I’ll tell you some consequences of such a model structure existing. It turns out
that everything has a relatively nice interpretation when we force this structure on a model category.
Proposition. If (C, w(C), I, J ) is a cofibrantly generated model category
(1) cof(C) = I-cof
(2) Every cofibration is a retract of a relative I-cell complex.
(3) The domains of I are small relative cofibrations
The same statements hold when cofibration is replaced by “trivial cofibration” and I replaced by J.
Theorem. Let C be a categories with subcategories w(C), I, J. Then (C, w(C), I, J ) forms a cofibrantly generated
model category if
(1)
(2)
(3)
(4)
(5)
w(C) has the two out of 3 property.
The domains of I and J are small relative I-cell and J-cell respectively.
J-cell ⊆ w(C) ∩ I-cof
I-inj ⊆ w(C) ∩ J-inj
Either wC ∩ I-cof ⊂ J-cof OR wC ∩ J-inj ⊆ I-inj.
We now come to the main examples. Always keep these in your brain.
Theorem. ChQ has a cofibrantly generated model structure given as follows. We let Sn be the chain complex with
a Q in degree n and nothing else. Define D n to be the chain complex with Qs in degrees n and n + 1 and the
differential dn : Q → Q the identity.
• The generating cofibrations, I, is the set the maps Sn−1 → D n .
• The generating fibrations, J, is the set of maps 0 → D n .
• The weak equivalences are quasi-isomorphisms.
Furthermore, the fibrations are surjections and the cofibrations are retracts of relative I-cell complexes.
Exercise. Look up a proof of this fact. One can be found in [?]. Convince yourself that chain homotopies,
which should be an old friend, crop up. Note that everything is fibrant.
Of course, topological spaces are also a model category, and in fact they are cofibrantly generated.
54
Theorem. Let Top be the category of topological space. A weak equivalence will be a usual sort of weak equivalence,
i.e. X → Y is a weak equivalence if π∗ ( X ) → π∗ (Y ) is an isomorphism. Let
I = { S n → D n : n ≥ 0}
J = { D n → D n × I : n ≥ 0}
This gives a cofibrantly generated model structure. Cofibrations are I-cof and fibrations are J-inj.
This is great! This is telling is that topological space in fact fit into a much more general framework for
doing homotopy theory. Some properties and pathologies of topological spaces are now explain in wider
setting.
Now, for the most part, even when you have a cofibrantly generated model category, proving the model
category axioms is a little annoying. However, there are ways to make our work easier in certain cases.
For example, if we happen to have adjunctions lying around where one member of the adjunction is a
model category, and the adjunctions play well with the model structure, then a model structure is induced
on the other category.
There is a general theorem for creating model structures out of adjunctions and cofibrantly generated
model categories. This will prove useful to us for the case of ChQ and CDGAs. The adjunctions that often
get used are free-forgetful adjucntions.
The following theorem is originally due to Kan. A proof can be found in [?].
Theorem. Suppose we have an adjunction F : C → D : U such that C has a cofibrantly generated model structure
( I, J ) and such that D satisfies the necessary closure axioms (i.e. closed under small limits and colimits). Suppose
• FI and FJ permit the small object argument
• U takes relative FJ-cell complexes to weak equivalences.
then there is a cofibrantly generated model category structure on D with ( FI, FJ ) as the requisite classes of maps.
The weak equivalences are maps that U takes to weak equivalences.
Here is the adjunction that will be useful for us:
Λ : ChQ CDGAQ : U
where given a chain complex C ∗ , Λ(C ∗ ) = ΛQ [Codd ] ⊗ Q[Ceven ]. The differentials are automatically
determined by this information and the fact that they must be differentials. From this adjuntion an the
theorem, we then have
Corollary. There is a model structure on CDGAQ such that
• The generating cofibrations are the maps ΛSn−1 → ΛD n
• The generating ayclic cofibrations are the maps 0 → ΛD n .
• The weak equivalences are quasi-isomorphisms.
Furthermore, the cofibrations are retracts of I-cell complexes and the fibrations are (degree-wise) surjections.
Example.
15.3. Homotopy Pushouts and Homotopy Pullbacks. We have
16. Appendix: Differential Graded Algebra
We’ll follow Larry Smith’s paper [?] for this, with some from Guggenheim-Munkholm thrown in.
17. Appendix: Hopf Algebras
Hopf algebras have an absurd number of conditions that they have to satisfy. But it’s easy to remember
most of them because it’s what the homology ring H∗ (ΩX; Z) satisfies.
References
1. Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354 (2002k:55001)
2. J. P. May and K. Ponto, More concise algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL,
2012, Localization, completion, and model categories. MR 2884233 (2012k:55001)
55