CGP DERIVED SEMINAR
GABRIEL C. DRUMMOND-COLE
1. April 4: Tae-Su Kim: morphisms of DG categories
I’m going to talk about morphisms of dg categories. These are topics in Bertrand
To¨en’s lecture notes. I’ll also talk about the homotopy category and about quasi-
equivalences and the homotopy category of dg categories.
I’ll give the definitions and examples and no theorems. So k will be a commuta-
tive unital ring.
Definition 1.1. A morphism of dg categories, or maybe I should call it a dg functor
between dg categories T and T
′
is a functor in the usual sense such that f, the map
between morphism spaces, T (x, y) → T
′
(f(x), f(y)), is required to be a chain map.
Let me give an example. Fix an object x of T and define a functor from
T → Ch(k), the chain complexes of k-modules. On objects it, takes y to T (x, y).
For morphisms it should take T (y, z) → Hom
Ch(k)
(T (x, y), T (x, z)) and here the
definition is obvious, it takes α to φ
α
which takes β to β ○ α.
We can check that dα goes to φ
dα
which maps β to β ○ dα. As Yoosik said,
dφ
α
(β) = ±φ
α
dβ ± d(φ
α
)β and these are the same by the compatibility condition
for the differential with composition.
Let me give another example. Let R asd S be k-algebras, unital associative. Let
f be a k-algebra morphisms from R to S.
Then we can define two functors f
∗
∶ C(R) → C(S) which are (S ⊗
R
−−) and f
∗
from C(S) → C(R), extension and restriction of scalars.
This might be a digression but let me tell you about the product between two
dg categories, the tensor product. Before doing that let me mention dgCat, which
has objects dg categories and morphisms dg functors. This forms a category.
Now let me take the tensor product of T and T
′
, this will be a dg category. The
objects are pairs of objects, one in T and one in T
′
. The morphisms Hom
T ⊗T
′
(x ⊗
x
′
, y ⊗ y
′
) is T (x, x
′
) ⊗ T
′
(y, y
′
).
My last example is a functor T ⊗ T
op
to Ch(k), and the objects (x, y) goes to
Hom
T
(y, x) and Hom
T ⊗T
op
((x, y), (x
′
, y
′
)) goes from Ch(k(T (y, x), T (y
′
, x
′
))) and
this takes γ to β ○ γ ○ α.
Let me move onto the second topic, the homotopy category of a dg category. We
can define a linear category [T ] and Yoosik gave a definiiton, the objects are the
same and the morphisms are H
0
(T (X, Y )). This is the “homotopy category of a dg
category” and this is a functor in the usual sense from dg Cat → Cat which takes
(T → T
′
) ↦ ([T ] → [T
′
]).
One issue in the homotopy category is composition H
0
(T (x, y))×H
0
(T (y, z)) →
H
0
(T (y, z)). [some discussion].
For C a linear category, then the homotopy category [C] concentreted on the
zero level and is isomorphic to C itself.
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