CGP DERIVED SEMINAR

GABRIEL C. DRUMMOND-COLE

1. Byung Hee An: Quillen adjunctions and equivalences

First of all it was hard to prepare this talk because I’m very much a beginner at

this category theory. If I say something wrong, then it’s Damien’s fault.

Let me say something about Quillen adjunctions and equivalences. Let’s start

with two categories C and D, we have two functors F ∶ C → D left adjoint to

G ∶ D → C.

Suppose both C and D have model category structures. In other words, C has

three classes of morphisms, weak equivalences and coﬁbrations and ﬁbrations, and

likewise for D . How can we say that these two model structures are related by

adjoint functors, by these two, right? So, to say about these two model categories,

related to those two adjoint functors, I want to ﬁrst state one lemma.

Lemma 1.1. (1) Suppose that F preserves coﬁbrations, that’s equivalent to G

preserving trivial ﬁbrations.

(2) The functor F preserves trivial coﬁbrations if and only if G preserves ﬁbra-

tions.

Let me give the idea of the proof. This is very easy. Let’s consider f ∶ X → Y

a C-coﬁbration. What does it mean to say that F preserves coﬁbrations? That

means that F (f) is in C

Consider a diagram like this:



G(A)

G(g )



G(B)

Then using the adjunction we have a diagram like this:

F (X)

F (f)



F (Y )

So if F preserves coﬁbrations, there is a lift of the second diagram, and then its

adjoint is a lift for the ﬁrst diagram. So then G(g) is in the right orthogonal of f,

and so is in the trivial ﬁbrations.

So among these four conditions, we pick two of these, like preserving coﬁbrations

and trivial coﬁbrations. So there are four equivalent conditions.

(1) F (C

) ⊂ C

F (W

∩ C

) ⊂ W

(2) ⋯

2 GABRIEL C. DRUMMOND-COLE

(3) ⋯

(4) ⋯

Deﬁnition 1.1. We call (F, G) a Quillen adjunction if one of the four equivalent

conditions is satisﬁed.

This doesn’t literally map a model structure to the other but it maps enough of

the structure of one to the other. So we call in this case F the left Quillen functor

and G the right Quillen functor.

Then if you have a Quillen adjunction, one of the nice properties is that those

functors induce functors at the level of the homotopy category. Before seeing that

I want to mention one remark: a left Quillen functor preserves weak equivalences

between coﬁbrant objects and a right Quillen functor preserves weak equivalences

between ﬁbrant objects.

This follows from “Ken Brown’s lemma” which Cheolgyu already mentioned. We

know that F preserves trivial coﬁbrations and then this is exactly the conclusion

that we draw from that lemma.

So now let’s consider a subcategory C

, the full subcategory whose objects are

coﬁbrants. Then F preserves coﬁbrants and this maps to D

, the full subcategory of

coﬁbrants. Now we take a localization to take a homotopy category, D

→ Ho(D

)

but this is the same as Ho(D). Then the composition satisﬁes a universal property,

that it maps all weak equivalences to isomorphisms, and this is the universal prop-

erty of the homotopy category, so it must factor through the homotopy category

of C

which is isomorphic to the homotopy category of C. So a left Quillen functor

induces a functor between homotopy categories, this is unique up to unique natural

transformation.

This induced functor we denote by LF . Similarly we can think the induced

functor from G, but instead of considering the coﬁbrants we think of the ﬁbrants,

we have D

, and since G preserves ﬁbrations this maps to C

, and then this localizes

to Ho(C), and this takes weak equivalences to isomorphisms so it factors through

Ho(D

) which is Ho(D), called RG.

We can consider C

, the full subcategory of coﬁbrant ﬁbrant objects, which sit

inside C

and C

, which sit inside C. These may not be model categories. But these

are categories with weak equivalences. These have a special class of morphisms with

the two out of three condition. Then we can make a homotopy category. Indeed we

have functors among all of these. The key point is that both satisfy the universal

property.

Sometimes these derived functors may be equivalences of categories. There are

several equivalent conditions for those derived functors to be equivalences of cate-

gories.

Lemma 1.2. The following are equivalent:

(1) The left derived functor LF is an equivalence

(2) the right derived functor RG is an equivalence.

(3) for any coﬁbrant X in C and ﬁbrant Y in D, then f ∈ Hom

(X, G(Y )) is

a weak equivalence if and only if its adjoint in Hom

(F (X), Y ) is a weak

equivalence

(4) For any coﬁbrant X, the composition X → GF X → G(R(F X)) (where R is

a ﬁbrant replacement) and for any ﬁbrant Y , the composition F (Q(GY )) →

F GY → Y is a weak equivalence.

DERIVED SEMINAR 3

These four conditions are equivalent.

Let me mention a weaker fact about the derived functors. So we have functors

LF ∶ Ho(C) → Ho(D) and RG ∶ Ho(D) → Ho(C), and these two functors are adjoint

to one another. The only thing to prove to see this is that F and G preserve some

homotopy relation. That’s basically about preserving cylinders or path objects.

But a cylinder is a factorization X ⊔ X → C(X) → X. Anyway, you can do it this

way or also with units and counits.

Let’s go back to the lemma. The ﬁrst two are equivalent because they are adjoint

to one another. It’s not too hard to prove the equivalences of the other statements,

but it’s not trivial. By the way, if you ﬁnd a Quillen adjunctions and Quillen

equivalences from google, then you can ﬁnd a website, the nlab, which says that

these last two are separate conditions that are equivalent, but that’s wrong.

I want to say only one sketch of the proof of only one thing because I think I’m

faster than I expected. I’m going to show one proof. Let’s see that the third and

fourth conditions are equivalent. The equivalence between the second and third

is in higher topos theory but not this one. So consider f ∶ X → GY , here X is

coﬁbrant and Y is ﬁbrant, then by adjunction there is a corresponding morphism

′

∶ F X → Y . and we apply ﬁbrant replacement and get a morphism

′

∶ RF X → Y .

But these two objects are ﬁbrant and so we take a functor G. We get

GF X



G(RF X)

So if we assume the fourth statement, then the composition along the top is a weak

equivalence. So by the two out of three condition, oh, suppose f

′

is a weak equiv-

alence. Since ﬁbrant replacement is weak equivalence, so is

′

. Then G preserves

weak equivalence between ﬁbrants so the arrow G(

′

) is as well. Then the compo-

sition is, so f is as well. If we use the other diagram we get the other direction.

Deﬁnition 1.2. A pair (F, G) is a Quillen equivalence if one of the conditions of

the lemma is satisﬁed.

I have these units and counits 1

Ho C

→ RG ○ LF , and being an equivalence means

that this is an isomorphism. Let’s assume that c is coﬁbrant, then what is LF (c)?

At the object level, the localization does nothing, so LF maps c to Ho(D), then

the image is a coﬁbrant thing. So then RG does a ﬁbrant replacement and takes

G. So the composition is something like C ↦ F (C) → D → G(D) where D is

ﬁbrant. If we pick any weak equivalence F (C) → D, the natural transformation

deﬁnes an isomorphism; then the composition C → G(D) we want to have as a

weak equivalence [missed something].

Let’s see some examples. I need some model structures to show examples. I

won’t go into any detail.

Let’s consider the category of topological spaces, with objects topological spaces

and morphisms continuous maps. As in chain complexes there are two standard

model structures, the “Quillen” model structure and the “Strøm” model structure.

In the ﬁrst one, the weak equivalences are weak homotopy equivalences, continuous

maps that induce isomorphisms on all homotopy groups. In the second model,

weak equivalences are homotopy equivalences. These two model structures cannot

be “the same” in some sense. If we localize and see homotopy categories then

4 GABRIEL C. DRUMMOND-COLE

these are diﬀerent. The identity functor from Top

→ Top

, this is not a Quillen

equivalence. This is a Quillen adjunction even though it’s not an equivalence.

Quillen to Strøm is the left adjoint.

The second example is the category of simplicial sets. This has objects functors

from ∆

to the category of sets. This is a “simplicial set” which has some kind

of face and degeneracy maps. Then this has a standard model structure. The

second example is that Top

and sSet are Quillen equivalent via ∣ ⋅ ∣, the geometric

realization, and the singular functor Sing, maps from the simplex. The important

thing is that this pair is a Quillen equivalence. Their homotopy categories are

equivalent. So if you only want to see homotopy types, in this category, then it’s

enough to consider simplicial sets. Damien said that the beneﬁt of simplicial sets

is that this is a presentable category.

In the Strøm category, coﬁbrations and ﬁbrations are maps satisfying the ho-

motopy extension and homotopy lifting properties, respectively. For the Quillen

case, it’s Serre ﬁbrations, which satisfy a lifting property only with respect to some

certain kind of maps D

→ D

× I.

I didn’t remark about coﬁbrant generation, where you have a small set of generat-

ing coﬁbrations, and then you can get all coﬁbrations from transﬁnite compositions,

pushouts, and retracts. You can do some kind of small object argument and build

some Serre coﬁbrations here.

A third example is chain complexes. We saw two model structures, but if A

and B are Abelian categories (with enough injectives), then we can make Ch(A)

and Ch(B), the chain complexes on them. Suppose that A and B are adjoint, then

there are induced functors on the chain complexes which are Quillen adjunctions.

The next example, I want to introduce one more model category, the category

sMod

, where this is simplicial R-modules. The deﬁnition is similar: functors

∆

→ Mod

. You have special maps, face and degeneracy. So the fourth ex-

ample is the Dold–Kan correspondence. Consider Ch(R) and sMod

, and there

are equivalences of categories, N and Γ which I don’t want to deﬁne. These are

not only just Quillen equivalences but also equivalences of categories. So, I missed

something. We have Quillen functors from Top to sSet and sMod

to Ch(R), and

we also have functors sMod

⇆ sSet. Maybe you remember the forgetful and free

functor between groups and sets. This is a forgetful functor, get rid of the module

structure, and in the other way it’s the free R-module. Then this pair is a Quillen

adjunction.

So I want to write in this way.

Top

⇆ sSet ⇆ sMod

⇆ Ch(R)

So here we have as composition the singular chain complex on spaces, so you get

homology at the level of homotopy categories.

I want to mention one more thing, this is not an example, say one more thing,

about transfered model structures. So suppose we have adjoint functors F ∶ C ⇆

D ∶ G, and suppose that C has a model structure. Can we deﬁne a model structure

on D using this adjunction? How can we hope to do this? The baby example is

when F and G are equivalences. Then it’s easy to get the model structure over to

the other place. You let the image be the desired class of morphisms. You only

have an adjoint pair. So what we want to do is to deﬁne the weak equivalences in

D to be G

−1

), all morphisms whose image under G is a weak equivalence, and

ﬁbrations all the maps whose preimage are ﬁbrations. When do you get this kind

DERIVED SEMINAR 5

of model structure? I don’t know a necessary and suﬃcient condition but I know a

suﬃcient condition.

Proposition 1.1. Assume that C is coﬁbrantly generated (I don’t want to deﬁne

this). Then (W

, F

) deﬁnes a model structure on D if it satisﬁes two condi-

tions (quite technical, I think. I’ll use stronger but easier conditions than the most

technical ones I know).

(1) G preserves ﬁltered colimits.

(2) D has a ﬁbrant replacement functor and path objects which are functorial

for ﬁbrant objects.

The ﬁrst condition we all know, you send a ﬁltered colimit to a ﬁltered colimit.

For the second one, we already deﬁned the ﬁbrations. So we don’t know the data

being from a model structure but we can still talk about ﬁbrant replacement.

If we decompose a morphism in that way, we can deﬁne a “path object” which

is a factorization of the diagonal A → P (A) → A × A, where the ﬁrst map is a

weak equivalence and the second a ﬁbration. We should be able to ﬁnd P (A) in a

consistent way.

The proposition says that if G and D satisfy these conditions then this data

deﬁnes a model structure on D.

Let’s see the example of transferred model structure. So for example sMod

can

be viewed as being transferred from sSet. So we can look at dg Alg

and Ch(R).

If you forget multiplication you get a chain complex.

I want to use the projective model structure, ﬁbrations are degreewise epimor-

phisms. Then there’s a canonical functor, the forgetful functor, and the other

way is a free functor. More concretely this is the free tensor algebra functor. So

T A =

⊕

⊗i

. Then the multiplication is tensor product. We don’t know the model

structure on dg algebras, but we have a model structure on Ch

and adjoint func-

tors. So we want to deﬁne weak equivalences and ﬁbrations as weak equivalences

and ﬁbrations under the forgetful functor. So these are algebra morphisms so that f

is a weak equivalence as chain complexes. It induces an isomorphism on homology.

Then ﬁbrations are degreewise epimorphisms.

The condition of the proposition is that

(1) forget preserves ﬁltered colimits

(2) the category of dg algebras should have ﬁbrant replacement and functorial

path objects for ﬁbrant objects.

The ﬁrst condition is true, even one hour ago I didn’t know why. Damien let me

know. Our ﬁbrations are degreewise epimorphisms. Any map from a dg algebra

to zero is a ﬁbration. So every element is ﬁbrant and the identity is a ﬁbrant

replacement functor. What about path objects? This is the situation in which the

proposition works. So I want a path object, which is actually quite complicated.

This can actually be deﬁned as A ⊗

Ω

poly

(∆

) if R is characteristic zero. Under

these certain assumptions, the path object can be written in this form, and then

it’s obvious that this is functorial. So this is the case.