CGP DERIVED SEMINAR

GABRIEL C. DRUMMOND-COLE

1. July 18: Byunghee An: Localization of dg categories

I’ll talk about localization of dg categories. So what’s the localization? Let T

be a dg category. We consider [T ] the homotopy category of T and let S be a

subset of the morphisms in [T ]. So then our goal is to deﬁne a localization of T

along S, another dg category L

T . This should satisfy some condition, regarding

morphisms contained in S as isomorphisms in some sense. This localization in some

sense is a dg localization. What does that mean? Consider a functor, sorry, okay,

from section two, we consider some functor F

T,S

from Ho(dg Cat) → Ho(Cat). If

we have a dg category T

′

then the target category is something F

T,S

′

) ⊂ [T, T

′

Basically it is a collection of dg functors up to homotopy, and f is in F

T,S

′

) if

and only if the induced functor [f] ∶ [T ] → [T

′

] sends S to isomorphisms in T

′

So then L

T is a dg category such that for any T

′

, [L

T, T

′

] is a subcategory

of [T, T

′

] and [L

T, T

′

] is F

T,S

′

Deﬁne a functor ` ∶ T → L

T . We call ` a localization of T along S if it

satisﬁes some universal condition. For any f ∶ T → T

′

such that [f](S) lives in the

isomorphisms of T

′

, then f factors through `, unique up to natural isomorphism.

Proposition 1.1. For any dg category T and for any S in the morphisms of [T ]

there exists a localization ` ∶ T → L

T .

So ` is actually a morphism in Ho(dg Cat).

Before showing the proof, I want to give an easy example. Recall the category

1 with one object and k as self-morphisms. Then ∆

has two objects 0 and 1 and

a distinguished morphism u from 0 to 1.

What if T is ∆

? A very simple example. T is [T ] and S = {u}. I’ll construct

` ∶ ∆

→ 1. This sends both objects to ∗ and this functor is obvious. I claim that

` is a localization of T along S. In other words it satsiﬁes the universal condition



′

The universal condition is equivalent to the fact that the `

∗

∶ [L

T, T

′

] → [T, T

′

] is

injective and the image of `

∗

is functors f which take S to isomorphisms.

The induced functor `

∗

∶ [1, T

′

] → [∆

, T

′

] is by precomposition. These were

computed in a previous exercise. We have [1, T

′

] ≅ Iso(T

′

), the isomorphism classes

of objects of T

′

. On the other hand, [∆

, T

′

] is the same as isomorphism classes

2 GABRIEL C. DRUMMOND-COLE

of morphisms in [T

′

]. If you have a morphism in [T

′

], the isomorphism class

means that you can pre- and post-compose with isomorphisms. This functor sends

the isomorphism class of an object to the class of the identity on that object,

[x] ↦ [Id

]. This is obviously injective.

For the second condition, let’s see what the isomorphism condition is for this

situation. Let f be a morphism in [∆

, T

′

] so [f ] is the same as a morphism

∆

→ [T

′

] and the condition is that [f](u) is an isomorphism. The image consists

of functors in which u is taken to an identity morphism, and thus an isomorphism.

Then the image of `

∗

is a functor which takes u to an isomorphism.

This is a simple example. Let me prove the existence of the localization in the

general case. I want to deﬁne L

T as a homotopy pushout

∐

∆



∐

So let S actually be the set of representing morphisms. We have a canonical mor-

phism ∆

→ 1, the canonical one, and we also have the map ∆

→ T taking u to

the morphism s.

I’ll give a deﬁnition of a homotopy colimit. Let D be a diagram like this, a

category, then we can consider C a model category and we can consider the diagram

category C

, and we can take the colimit C

→ C. We can construct a functor the

other way by a constant functor. Then actually these two are adjoint to each other

and are a Quillen adjunctions so induce functors on the homotopy category. Then

the left derived functor of the colimit is the homotopy colimit. If D is just the

pushout diagram, then this is just the homotopy pushout.

Now we have an explicit deﬁnition of the localization. So how to see that this

is a localization? We have the map from T to L

T from it being a (homotopy)

pushout. How about the universal property? Let’s consider T

′

? Then f sends all

morphisms f to isomorphisms.

Before we have a break, I want to give an exercise. Show that L

∆

is equivalent

to 1. So this is saying that 1 ⊔

∆

but this is a coﬁbrant diagram (trust me on

this), and this is just the regular pushout, but pushing out along the identity gives

the other object.

Okay, the next thing I want to talk is an exercise. Let T and T

′

be two dg

categories and S and S

′

collections of morphisms in the homotopy categories of T

and T

′

as usual, containing all identities. Then the derived tensor L

T ⊗

′

equivalent to L

S⊗

′

T ⊗

′

in the homotopy category of dg categories. We know

T ⊗

′

but we need to deﬁne S ⊗

′

What is the derived tensor product T ⊗

′

? I erased the deﬁnition. This is, in

our paper, this is Q(T ) ⊗ Q(T

′

), where Q is a coﬁbrant replacement functor. So

what’s the right deﬁnition of S ⊗

′

? This is a set of morphisms in the homotopy

categories of morphisms. We can safely use the representing morphisms. We can

in some sense remove the brackets. Then these, I want to say something like

Q(S) ⊗ Q(S

′

), this is not deﬁned but let me write this, consider s ∶ x → y in S. If

we take Q, then we get Qx → x and Qy → y. We want S ⊗

′

⊂ Mor([T ⊗

′

]).

How to prove the exercise? I think I’m wrong, but let’s look at the right hand

side. Let M be a C(k)-model category. Look at [L

S⊗

′

(T ⊗

′

∫

(M)], this sits

DERIVED SEMINAR 3

in [T ⊗

′

∫

(M)], this should be injective and the image has some property. We

have injectivity here, and by using something we know, by using the universal prop-

erty of the internal homomorphisms, we can move this to [T, RHom(T

′

∫

(M))]

which is the same as [T,

∫

′

)] which is the same as Iso(Ho((M

′

)

)), and then

we can switch it around, this is the same as Iso(Ho(M

T ⊗T

′

)), and what I’ve done

is gotten rid of L.

[some discussion of alternate methods]

So instead let’s take ` ⊗

′

∶ T ⊗

′

→ L

T ⊗

′

. So suppose we have a

functor F to T

′′

and suppose it takes morphisms in S ⊗

′

to isomorphisms, then

we want to lift to L

T ⊗

′

[some more discussion]

The next one is the following proposition.

Proposition 1.2. Let M be a coﬁbrantly generated C(k)-model category and M is

M viewed as a dg category. Then there is an isomorphism in Ho(dg Cat) Int(M) ≅

If we invert every weak equivalence in a dg sense, then we get Int(M).

The proof is actually not hard, relatively easier than this exercise. We have a

functor Int(M) → M

→ M, the Int(M) is coﬁbrant and ﬁbrant objects, and M

is the ﬁbrant objects. We denote the inclusions

Int(M)

;;

and we can deﬁne ﬁbrant replacement r and coﬁbrant replacement q. The existence

of these functors q and r comes from coﬁbrant generation of M (but you could just

assume them). Then (q ○ j)(x) = q(x) → x and (j ○ q)(x) is again q(x), what I

mean is there are natural transformations (q ○ j) → id and (j ○ q) → id and then

id → r ○ k and id → k ○ r. These are natural weak equivalences.

Then there are isomorphisms L

Int(M) → L

→ L

M in Ho(dg Cat). So

what we want to prove is that L

(M) ≅ Int(M), and my claim is that L

(Int(M)) ≅

Int(M). Localization means that we want to invert some morphisms in M. We

want to declare certain morphisms to be isomorphisms. But W was already, well,

[Int(M)] ≅ Ho(M) ≅ M[W

−1

]. By the deﬁnition of localization we have a functor

Int(M)

Ð→ L

Int(M). Then if we have f ∶ Int M → T

′

satisfying some condition

then we get a lift from L

Int(M). But what’s the condition? It’s that when we

have [Int(M)] → [T

′

] it sends W to isomorphisms. But this condition is vacuous.

So there’s a correspondence.

The next one is the last one:

Proposition 1.3. Let T be a dg-category and S some subset of Mor([T ]), then

` ∶ T → L

T induces a functor `

∗

∶ D(L

(T )) → D(T ) which is fully faithful and so

that F ∶ T → C(k) is in im`

∗

if and only if F (S) lands in quasi-isomorphisms.

Proof. D(T )

∶

= Ho(T − mod ) and D(L

T )

∶

= Ho(L

T − mod ), and this is the

same as [Int(C(k)

)] which in turn is the same as [RHom(L

T, Int(C(k)))].

We are comparing this to [RHom(T, Int(C(k)))]. Then the functor is well-deﬁned

by functoriality of RHom. To prove fully faithfulness, we ﬁrst consider the functor

4 GABRIEL C. DRUMMOND-COLE

without brackets, `

∗

from RHom(L

T, Int(C(k))) ≅

∫

(C(k)

) ⊂ L

T − mod to

RHom(T,

∫

(C(k))) ≅

∫

(C(k)

) ⊂ T − mod . We have `

∗

from L

T − mod → T −

mod , and fully faithfulness there (or the quasi-version) should imply it at the lower

level. But that I didn’t show.

The second condition is easy. The image of `

∗

are the ones that factor through

T , and so something in S, they must go to quasi-isomorphisms in order to

be isomorphisms in the homotopy category. I couldn’t show the proof for full

faithfulness. But I think this is not too hard. 

I want to introduce one last exercise without an answer. The ﬁnal exercise is

like this.

Let ` be the T → L

T a localization of dg categories. Then we can consider,

this induces, f ∶ T → T

′

induces M

← M

′

, this is natural because the objects are

functors and we can compose with f . But actually there is a left adjoint f

, here

M is a C(k)-model category. This pair is a Quillen adjunction and moreover if f

is an equivalence (isomorphism in the homotopy category) then these are a Quillen

equivalence. This part is Lemma 1, but it uses this fact about f

. Especially if f is a

localization, then the left adjoint `

exists and its derived functor L`

, well, ﬁrst `

∶

T − mod → L

T − mod , then the derived functor is on the homotopy categories.

So this is D(T

) → D(L

). We’ve got a functor between derived categories.

Let W

be the morphisms in D(T

) such that L`

(u) is an isomorphism. The

ﬁrst question is about classifying the elements of W

. I don’t want to talk about

it. The second is more interesting. We can invert W

(which satisﬁes a 2 out of 3

property in D (T

)) and the second statement is that D(T

)[W

−1

]

Ð→ D(L

)

and that’s an equivalence of categories. So you can localize either before or after

passing to modules.

I wanted to solve this but I don’t have enough time. I’ll stop.