LECTURE 7: MODEL CATEGORIES, II

We begin by covering some more basic categorical facts relating to model categories. By definition, a cobase change of a map f : X → Y in any , is a map g : W → Z that fits in pushout diagram

X W f g Y Z.

Similarly, a base change of a map f : X → Y is a map g : W → Z that sits in a pullback diagram

W X g f Z Y.

Proposition 0.1. Suppose C is a . Then: (1) Cof ⊂ Mor(C) is closed under cobase change. (2) Cof ∩ W ⊂ Mor(C) is closed under cobase change. (3) Fib ⊂ Mor(C) is closed under base change. (4) Fib ∩ W ⊂ Mor(C) is closed under base change.

Proof. We prove (1), the rest are similar. Let f : X → Y be a cofibration and g : W → Z be a cobase change of f. By the axioms of a model category, it is enough to show that g has the LLP with respect to acyclic fibrations. Indeed, suppose p : A → B is an acyclic fibration and consider a diagram

W A g p Z B.

We want to produce a h : Z → A. Amending this diagram to the left with f we have a diagram

X W A f p Y Z B.

The dotted arrow lift exists since f is a cofibration. By the of pushouts, the dotted arrow defines a map h : Z → A lifting g against p as desired.  1 1. Localization for model categories

Localization is a general notion one can define for any category with weak equivalences. For model categories, we see that localizations exists in a constructive way. The key idea is that one can use the concept of equivalence, which makes sense in a general model category. We met the definition of homotopy equivalence for general maps in a model category in the last lecture. We begin by exhibiting a class of objects for which homotopy equivalences are particularly well-behaved.

1.1. (Co)Fibrancy. Throughout, C is a model category.

Definition 1.1. An object X in a model category C is called cofibrant if the unique map 0 → X is a cofibration. An object X in a model category C is called fibrant if the unique map X → ? is a fibration.

Remark 1.2. Every object X admits a cofibrant replacement

0 X ' Cof Fib∩W QX and a fibrant replacement: X ? ' Cof∩W Fib RX .

When we restrict ourselves to maps from a cofibrant object, we see that left homotopy equivalence is actually an equivalence relation!

Proposition 1.3. Suppose X is cofibrant. Then, left homotopy equivalence ∼L is an equivalence relation on the set HomC(X,Y ).

Before proving this, we prove the following lemma.

Lemma 1.4. Suppose X is cofibrant, and let Cyl(X) be a cylinder object. Then, the maps

j1,j2 i1, i2 : X −−−→ X t X → Cyl(X) are both acyclic cofibrations.

Proof. Let’s prove this for i1. By assumption, we can factor idX : X → X as

X −→i1 Cyl(X) −→' X.

By two out of three, we know that i must be a weak equivalence. (This does not yet use cofibrancy.) 2 Now, X t X is a , so it fits in a pushout diagram 0 X

j1 j X 2 X t X. By assumption 0 → X is a cofibration. Since cofibrations are closed under cobase change, we conclude that j1 : X → X t X is a cofibration. Finally, cofibrations are closed under composition, we see that i1 is a cofibration.  Now on to the proof of the proposition.

Proof. Since X is cofibrant, we can take A to be its own cylinder object Cyl(X) = X. Then, we see that any map f : X → Y is left homotopy equivalent to itself using homotopy f : Cyl(X) = X → X. Thus f ∼L f via f.

Now, we want to see that f ∼L g then g ∼L f. If σ : X t X → X t X is the flip map, then we have (f + g) ◦ σ = g + f. 0 Finally, suppose f ∼L g and g ∼L h with left H : Cyl(X) → X, H : Cyl(X)0 → X. Then, consider the pushout diagram

i0 X ' Cyl(X) 0 i1 ' Cyl(X)0 Z

0 By the lemma above, we know that i0, i1 are acyclic cofibrations. Thus, by the universal property of pushouts it is immediate to see that Z itself is a cylinder object for X, Z = 00 00 00 Cyl(X) and the induced map H : Cyl(X) → X is a left homotopy f ∼L h.  A very similar proof shows the dual notion.

Proposition 1.5. Suppose Y is fibrant. Then ∼R is an equivalence relation on HomC(X,Y ).

In fact, if X is also cofibrant, then f, g ∈ HomC(X,Y ) satisfy

f ∼L g ⇐⇒ f ∼R g. Thus, when we look at the full of objects that are both fibrant and cofibrant

C◦ ⊂ C the relations ∼L, ∼R are equivalent equivalence relations. Definition 1.6. The hoC of a model category C is the subcategory of C consisting of objects that are fibrant and cofibrant (so the same objects as C◦), and are

HomhoC(X,Y ) = HomC(X,Y )/ ∼ where ∼ is either ∼L or ∼R. 3 1.2. Back to localization. We go back to the notion of localization of a category with weak equivalences. Given any model category C, we will see that hoC is a model for its localization.

Proposition 1.7 (Whitehead). Suppose C is a model category and X,Y are both fibrant and cofibrant. Then, f : X −→' Y is a weak equivalence if and only if f is a (left or right) homotopy equivalence.

Proof. First, the forward direction. By the factorization axioms of a model category we can factor any map as an acyclic cofibration followed by a fibration. If f is a weak equivalence, by the two out of three property we see that f actually factors as an acyclic cofibration followed by an acyclic fibration.

f X Y g h Cof∩W Fib∩W Z . Since X,Y are both fibrant and cofibrant, we see that Z is also both fibrant and cofibrant. Since the composition of homotopy equivalences is a homotopy equivalence, it suffices to show that both g, h are homotopy equivalences. We consider the acyclic fibration h : Z → Y , where both Z,Y are fibrant cofibrant. The proof for the acyclic cofibration g : X → Z is completely similar. By the LLP of cofibrations against acyclic fibrations, there exists a dotted map eh−1 in the diagram 0 Z h−1 Cof e h Z Z.

−1 −1 Note that eh is an ordinary right inverse to h, h ◦ eh = idZ . We want show that −1 eh ◦ h ∼L idZ . Let Cyl(Z) be a cylinder object for Z

Z t Z → Cyl(Z) −→r Z and consider the commuting diagram

h−1◦h+id Z t Ze ZZ

h Cyl(Z) h◦r Y.

Note the diagram commutes since eh−1 is a right inverse to h. Again, by the LLP of cofibrations against acyclic fibrations, the dotted map above exists. One immediately checks that this is the desired homotopy. Now, for the converse.  4 This proposition implies the following result of Whitehead.

Theorem 1.8 (Whitehead). Let C be a model category. Then, there is an equivalence of categories ∼ C[W−1] −→= hoC.

1.3. Another model for the homotopy category. There is another description of the homotopy category of a model category. We have already mentioned how every object X ∈ C admits a, non-unique, fibrant and cofibrant replacement that we denote by RX and QX, respectively. Although these are not unique replacements when viewed in the whole category, they do descend to

fib R :C → C / ∼R cof Q :C → C / ∼L .

fib Here, C / ∼R is the category of fibrant objects with morphisms given by ∼R-equivalence classes. Similarly for Ccof . If we restrict R to the full subcategory of cofibrant objects, we see that it induces a

cof ◦ R|cof : C → C / ∼= hoC.

Similarly, the functor Q restricted to Cfib determines a functor

fib ◦ Q|fib : C → C / ∼= hoC.

Proposition 1.9. The homotopy category hoC is equivalent to the category consisting of all objects of C, and whose morphisms between X,Y ∈ C are

HomhoC(R|cof QX, R|cof QX).

The equivalence is induced by the natural functor C → hoC.

Remark 1.10. We could have also used HomhoC(Q|fibRX,Q|fibRX) and this would give yet another equivalent category to hoC.

dg 2. Model structure on Vectk dg There is a model structure on the category C = Vectk of dg vector spaces over a field k where: • The weak equivalences are the quasi-. That is, maps of dg vector spaces f : V → W inducing and H∗f : H∗V → H∗W . • The cofibrations are maps of dg vector spaces f : V → W that are degreewise injective maps of vector spaces:

f n : V n ,→ W n. 5 • The fibrations are maps of dg vector spaces f : V → W that are degreewise surjective maps of vector spaces:

n n n f : V  W .

dg Theorem 2.1 (Hovey). The choices above make Vectk into a cofibrantly generated model category.

Roughly, cofibrantly generated means that there is a set cofibrations that generate all other cofibrations. We will come back to the concept later.

3. Adjunctions

A natural question is when a functor F : C → D between model categories descends to a functor between the associated homotopy categories. When F is a homotopy invariant functor, meaning it preserves weak equivalences, then it does indeed descend

F : hoC → hoD.

The problem is, homotopy invariance is a quite a strong condition, and many functors we run into in practice do not satisfy the condition. To answer the question in more generality, we introduce the concept of a . First, the following general idea.

Definition 3.1. A left of a functor F : C → D along a functor G : C → E is a pair of a functor LanG(F ): E → D fitting into the diagram

C F D

G LanG(F ) E together with a ηL : F → LanG(F ) ◦ G that is initial among all such pairs.

A right Kan extension of F along G is a pair of a functor RanG(F ): E → D fitting into the diagram C F D

G RanG(F ) E together with a natural transformation ηR : RanG(F )◦G → F that is final among all such pairs.

When Kan extensions exists they are unique by their universal properties. 6 Definition 3.2. Let C be a model category, D be any category, and F : C → D a functor. 0 A left derived functor L F of F is a right Kan extension of F along the C-localization ΓC : C → ho(C): 0 L F = RanΓC (F ) : hoC → D. 0 A right derived functor R F of F is a left Kan extension 0 R F = LanΓC (G) : hoC → D. When both C and D are model categories, there is a notion of a total derived functor.

Definition 3.3. Let C, D be model categories, and F : C → D a functor. A total left derived functor LF of F is a right Kan extension of the composition ΓD ◦ F along the C-localization ΓC : C → ho(C):

LF = RanΓC (ΓD ◦ F ) : hoC → hoD.

A right derived functor RF of F is a left Kan extension

RF = LanΓC (ΓD ◦ G) : hoC → hoD. Recall, we say a functor F : C → D is left adjoint to a functor U : D → C if there exists a natural isomorphism of bifunctors C × D → Set

HomD(F (−), −) → HomC(−,U(−)). Thus, for each X ∈ C and Y ∈ D we have =∼ HomD(F (X),Y ) −→ HomC(X,U(Y )). In symbols, we write F a G.

Definition 3.4. An adjunction between model categories

F C a D U is a if F preserves cofibrations and acyclic cofibrations.

Remark 3.5. Equivalently, a Quillen adjunction is an adjunction as above such that U that preserves fibrations and acyclic fibrations. For instance, note that all of the data in the diagram FX Y

Cof∩W Fib FX0 Y 0 is equivalent, by the adjunction, to the diagram X UY

Cof∩W Fib X0 UY 0. 7 Theorem 3.6 (Quillen). If F a U is a Quillen adjunction, then LF and RU exist and define an adjoint pair between homotopy categories

LF hoC a hoD

RU Furthermore, LF and RU are computed by formulas

LF (X) = ΓDF (QX)

RG(Y ) = ΓCG(RY ).

Remark 3.7. It is not necessary to assume F,U participate in a Quillen adjunction to ensure that LF, RU exist. However, their formulas may not be as simple as in the theorem.

Definition 3.8. A Quillen equivalence is a Quillen pair F a U such that LF a RU is an equivalence of respective homotopy categories.

Theorem 3.9 (Quillen). Suppose F a U is a Quillen pair. If, for every cofibrant object X ∈ Ccof and fibrant object Y ∈ Cfib one has

(f : X → UY ) ∈ WC ⇐⇒ (ηf : FX → Y ) ∈ WD then LF (and RG) defines an equivalence of categories

LF hoC ∼= hoD.

RU

8