Maps in the Homotopy Category of dg-Categories

CGP DERIVED SEMINAR

WANMIN LIU

1. July 4: Wanmin Liu (Maps in the Homotopy Category of

dg-Categories, part II)

Outline:

● recall the Proposition 4.1.2 which links the dg-side (morphisms) to model-

side (objects) (introduced by Tae-Su and proved by Gabriel on June 27).

● an exercise by using the Proposition.

● a corollary of the Proposition.

Recall that for any dg-category T , we also denote a T − Mod by (C(k))

Proposition 1.1. [To11, Prop 4.1.2] Let T be any dg-category and M be a C(k)-

model category satisfying technical conditions:

(1) M is coﬁbrantly generated, and the domain and codomain of the generating

coﬁbrations are coﬁbrant objects in M.

(2) For any coﬁbrant object X in M , and any quasi-isomorphism E Ð→ E

′

C(k), the induced morphism E ⊗ X Ð→ E

′

⊗ X is an equivalence.

(3) Inﬁnite sums preserve weak equivalences in M .

Then there exists a natural bijection of sets

[T, Int(M )] ≃ Iso(Ho(M

))

between the set of morphisms from T to Int(M) in Ho(dg − cat) and the set of

isomorphism classes of objects in Ho(M

Exercise 1.1. [To11, Exercise 4.1.6 or Exercise 21 in the preprint version] Let R be

an associative and unital k-algebra, which is also considered as dg-category with a

unique object and R as endomorphisms of this object. Show that there is a natural

bijection between [R, Int(C(k))] and the set of isomorphism classes of the derived

category D(R).

Proof. Let us recall some notions. We denote the associated dg-category of the R

by BR (notations after [To11, Deﬁnition 3.2.3] or the preprint version page 31).

More precisely, Obj(BR) is the unique object, denoted by ∗. To make it as a dg-

category, for any “two” object(s) ∗, and ∗, we need to deﬁne a chain complex of

k-module. The natural choice is

BR(∗, ∗) ∶= R ∈ C(k), i.e.⋯ → 0 → R → 0 → ⋯ ∈ C(k),

where R is in the 0-th degree.

So [R, Int(C(k))] really means [BR, Int(C(k))] as in the category Ho(dg−cat),

where the dg-category structure on Int(C(k)) is induced from the dg-category

C(k). More precisely, it is the full sub-dg-category of C(k) consisting of ﬁbrant

and coﬁbrant objects in C(k), where you need to recall the standard model category

structure on C(k).

2 WANMIN LIU

To use the Proposition 1.1 (Prop 1 of page 38 in preprint), we need to assume

that R is ﬂat over k (see also Exercise 4.1.1).

Now the Proposition gives us

[BR, Int(C(k)] ≃ Iso(Ho(C(k)

)).

By the Deﬁnition 3.2.3, D(BR) ∶= Ho(BR − Mod) = Ho(C(k)

To ﬁnish the proof, we only need to check

D(BR) ≃ D(R).

Then it follows that

[BR, Int(C(k)] ≃ Iso(Ho(C(k)

)) = Iso(D(BR)) ≃ Iso(D(R)).

Now let us check the equivalence relation D(BR) ≃ D(R).

What is D(R)? It is deﬁned as S

−1

C(R), the localizing of C(R) along S, where

C(R) is the category of chain complex of left R-modules, and S is the set of quasi-

isomorphisms. (See Yoosik’s lecture on Apr 4.)

What is D(BR)? It is deﬁned as W

−1

(BR − M od), the localizing of BR − Mod

along W , where BR −Mod is the model category, with the model structure induced

by the dg-structure of BR, and W is the set of weak equivalences, which is deﬁned

to be the set S of quasi-isomophisms. (See Morimichi ’s lecture on June 20.)

Claim. By forgetting the model structure on BR−M od, we have an equivalence

of categories BR − Mod ≃ C(R).

Let us check in the object level. An object in BR − M od is a dg-functor

F ∶ BR Ð→ C(k), which maps the unique object ∗ to F

∗

∈ C(k), together with

morphism in C(k)

∗

⊗

BR(∗, ∗) Ð→ F

∗

and for any “pair” of objects (∗, ∗) ∈ Obj(BR)

a morphism in C(k),

∗,∗

∶ BR(∗, ∗) Ð→ C(k)(F

∗

, F

∗

) = Hom

∗

, F

∗

satisfying the usual associativity (here we need R to be an associative k-algebra):

BR(∗, ∗) ⊗

BR(∗, ∗)

∗,∗

⊗F

∗,∗



∗,∗,∗

BR(∗, ∗)

∗,∗



Hom

∗

, F

∗

) ⊗ Hom

∗

, F

∗

)

′

∗,∗,∗

Hom

∗

, F

∗

and unit conditions (here we need R to be a unital k-algebra):

∗

′

∗

BR(∗, ∗)

∗,∗



Hom

∗

, F

∗

We therefore associate each object F ∈ Obj(BR − M od) a complex of R-modules

∗

⊗

BR(∗, ∗) ∈ Obj(C(R)).

Let us check in the morphisms level. A morphism in BR − M od is a natural

transformation between dg-functors α ∶ F Ð→ F

′

, which maps F

∗

Ð→ F

′

∗

with the

DERIVED SEMINAR 3

commutative diagram

∗

⊗

BR(∗, ∗)

∗

⊗

BR(∗,∗)



∗



′

∗

⊗

BR(∗, ∗)

′

∗

We therefore associate each morphism α ∈ Mor(BR − M od) a morphism α

∗

⊗

BR(∗, ∗) ∈ M or(C(R)). Moreover, α is deﬁned to be a weak equivalence in BR −

Mod (i.e. α ∈ W ) if α

∗

⊗

BR(∗, ∗) is a quasi-isomorphism in C(R) (i.e. α

∗

⊗

BR(∗, ∗) ∈ S).

Therefore by localizing the equivalence of categories BR−Mod ≃ C(R) along the

same class of quasi-isomorphisms, we obtain the equivalence of categories D(BR) ≃

D(R).



Corollary 1.1. [To11, Page 279 Corollary 1 or page 37 of preprint version] Let T

and T

′

be two dg-categories, one of them having coﬁbrant complexes of morphisms.

Then, there exists a natural bijection between [T, T

′

] and the subset of Iso(Ho(T ⊗

′

)

−Mod)) consisting of T ⊗(T

′

)

-dg-modules F such that for any x ∈ T , there

is y ∈ T

′

such that F

x,−

and h

are isomorphic in Ho((T

′

)

− Mod).

Proof. Let us denote the subset in the corollary by N . Let us denote the model

category (T

′

)

−Mod by M . We need to show the following commutative diagram:

[T, T

′

]





≃





[T, Int(M )]

≃

Iso(Ho(T ⊗ (T

′

)

− Mod)),

i.e. we need to check that those conditions on N as a subset of Iso(Ho(T ⊗(T

′

)

−

Mod)) exactly correspond to the left vertical inclusion, and the correspondence

from left to right is given by the Proposition 4.1.2.

Firstly, let us check the bottom equivalence in the diagram. Recall notations

M = (C(k))

′

)

, T ⊗ (T

′

)

− Mod = (C(k))

T ⊗(T

′

)

In Youngjin ’s lecture (Exercise 3.2.6), we know the equivalence of model categories

((C(k))

′

)

≃ (C(k))

T ⊗(T

′

)

The bottom equivalence follows by the Proposition 1.1.

Secondly, let us check the left inclusion in the diagram. Recall that the C(k)-

enriched Yoneda embedding is a quasi-fully faithful dg-functor h (in Tae-Su’s

lecture on June 27):

h ∶ T

′

Ð→ Int(M),

sending an objects y ∈ T

′

to an object h

in the interior of M . Here

∶ (T

′

)

Ð→ C(k), z ↦ T

′

(z, y).

The interior Int(M ) is deﬁned as the full subcategory of M consisting of ﬁbrant

and coﬁbrant objects of M . The quasi-fully faithful dg-functor means that the

induced morphism of complexes of k-modules

∀x, y ∈ T

′

, h

x,y

∶ T

′

(x, y) Ð→ Hom(h

, h

) = Int(M)(h

, h

)

4 WANMIN LIU

is an isomorphis. And the complex Hom(h

, h

) ∈ C(k) is deﬁned from the C(k)-

model structure on M by

Hom

C(k)

(E, Hom(h

, h

)) ≃ Hom

(E ⊗ h

, h

), ∀E ∈ C(k).

Recall that [T, T

′

] and [T, Int(M)] are morphisms in the category Ho(dg−cat). We

need to work in the category dg−cat ﬁrst. By Tabuada’s Theorem (in Yong-Geun ’s

lecture on June 12), there is a model category structure on dg − cat. By Lemma

4.1.3, we can assume that T is a coﬁbrant dg-category. Recall that all objects in

the category dg − cat are ﬁbrant. By restriction to the sub-category (dg − cat)

coﬁbrant and ﬁbrant objects, we hence get an equivalence of categories (here ∼ is

the homotopic equivalence):

(dg − cat)

/ ∼ Ð→ Ho(dg − cat).

In the (dg − cat)

, by composing h we obtain

Hom

(dg−cat)

(T, T

′

) ↪ Hom

(dg−cat)

(T, Int(M )),

which descends to (recall the notation [T, T

′

] ∶= Hom

Ho(dg−cat)

(T, T

′

))

[T, T

′

] ↪ [T, Int(M)],

whose image consists of morphisms in Hom

(dg−cat)

(T, Int(M )) factorizing in

Ho(dg − cat) through the quasi-essential image of h.

Finally, let check the diagram is commutative, i.e. for an object F in the right-

bottom and assume it comes from top-left by the left inclusion and bottom equiva-

lence, then F is the form as in the deﬁnition of N. Let F ∈ Obj(T ⊗ (T

′

)

− M od),

i.e. a dg-functor

F ∶ T ⊗ (T

′

)

Ð→ C(k).

By the bottom equivalence of the diagram, we regard F ∈ [T, Int(M)], i.e. F =

F / ∼

for some lift

F ∈ Hom

(dg−cat)

(T, Int(M )). To check the top equivalence of the

diagram, we further regard

F in the essential image of h. Therefore,

F ≃ h ○ g, for some g ∈ Hom

(dg−cat)

(T, T

′

)

For any x ∈ T , we just take

y ∶= g(x) ∈ T

′

Hence

F (x) ≃ h(y) = h

∈ M.

Since the lift

F is upto homotopy, we obtain F

x,−

∼

F (x) ∈ M and hence

x,−

≃ h

∈ Ho(M).



References

[To11] B. To¨en, Lectures on DG-Categories, Topics in Algebraic and Topological K-Theory, edited

by P.F. Baum et al., Lecture Notes in Math. 2008, Springer-Verlag Berlin Heidelberg, 2011.