CGP DERIVED SEMINAR
GABRIEL C. DRUMMOND-COLE
1. April 4: Yoosik Kim: DG categories
Last time we saw the definition of the derived category of an Abelian category,
first passing to the homotopy category and then inverting the quasi-isomorphisms.
This is not an Abelian category in general, but it’s triangulated, it has a shift functor
and some distinguished triangles to generate long exact sequences in homology.
This triangulated category is not the best structure, so what we’re going to do is
to enhance this category using the notion of dg categories. My mission is to define
dg categories and give some typical examples to get familiar with that.
So let k be a commutative ring, you can think of it as the ring of integers or a
field if you prefer linear algebra.
Definition 1.1. A k-linear category A is called a dg (differential graded) cate-
gory if the morphism spaces are dg k-modules and the compositions and units are
morphisms of dg k-modules.
I probably need to explain some terminology here. Recall that a category A is
said to be k-linear if the morphism spaces are k-modules, that is, Hom
A
(X, Y ) is
a k-module and the composition Hom
A
(Y, Z) ⊗ Hom
A
(X, Y ) → Hom
A
(X, Z) is a
k-module homomorphism. A dg k-module V is
(1) V =
V
p
(2) d
V
a differential, d
V
(V
p
) ⊂ V
p+1
.
A morphism f ∶ V → W of dg k-modules of degree n has f(V
p
) ⊂ W
p+n
and
d
W
○ f = (−1)
n
f ○ d
V
The tensor product V ⊗ W of graded k-modules has
(1)
(V ⊗ W )
p
=
i+j=p
(V
i
⊗ W
j
)
(2) d
V ⊗W
= d
W
⊗ id
W
+ id
V
⊗ d
W
.
The tensor product of morphisms of graded k-modules, f ∶ V → V
′
and g ∶ W → W
′
,
this is defined by the Koszul sign rule, (f ⊗ g)(v ⊗ w)
∶
= (−1)
∣g∣∣v∣
f(v) ⊗ g(w).
So a first example of a dg category is a dg algebra, a graded algebra over k, so
it has a grading, a multiplication, this is a dg category with one object, d ∶ A → A
satisfies the graded Leibniz rule d(a ⋅ a
′
) = da ⋅ a
′
+ (−1)
∣a∣
ada
′
.
Any dg algebra is a dg category with a single object. Conversely, a dg category
with a single object can be viewed as a dg algebra. Take {∗} as the objects and
Hom
A
(∗, ∗) = A, with composition from multiplication.
1
2 GABRIEL C. DRUMMOND-COLE
From this convention, you view this as a dg algebra, and the graded Leibniz rule
crmes from our convention,
d(a ⋅ a
′
) = µ [(d ⊗ id
A
+ id
A
⊗ d
A
)(a ⊗ a
′
)] = da ⋅ a
′
+ (−1)
∣a∣
a ⋅ da
′
The most important example, let A be a k-linear Abelian category. Then C(A)
the category of chain complexes in A. Then this can be made into a dg category as
follows. The objects and composition are the same as in C(A) and the morphisms
for chain complexes C and D, the gradings are Hom
Cdg
(C, D) =
p
Hom
Cdg
(C, D)
p
where Hom
Cdg
(C, D)
p
is
∏
i
Hom
A
(C
i
, D
i+p
).
Now I have to define the differential, so d
p
takes f
i
to d
i+p
D
○f
i
+(−1)
p+1
f
i+1
○d
i
C
.
So if I draw one piece this looks like
C
i
d
C
//
f
i
C
i+1
f
i+1
D
i
d
D
//
D
i+1
so that d measures the failure of commutativity of this diagram.
That’s the primary example of a dg category. You can take Z
0
, objects are the
same as in Cdg(A). The morphisms are the kernel, they’re the kernel of d
0
. Then
H
0
(Cdg(A)) has the same objects, and the morphisms are ker d
0
/imd
0
, which are
chain maps modulo homotopy.
For a later purpose you want the opposite category, if A is a dg category with
d and ○ then the opposite category A
op
consists of the following data, the objects
are the same as in A and the morphisms are opposite; the differential d
op
(X, Y ) =
d(Y, X) and the composition g ○
op
f is defined as (−1)
∣g ∣∣f ∣
f ○ g.
Now let me introduce A
∞
categories, which is a cousin of a dg category.
Definition 1.2. A (unital) A
∞
category A consists of objects, and morphisms
Hom
A
(X, Y ) a Z or Z/2-graded kmodule and it comes with a composition, structure
maps m
d
which goes
Hom
A
(X
0
, X
1
) ⊗ ⋯ ⊗ Hom
A
(X
d−1
, X
d
) → Hom
A
(X
0
⊗ X
d
)
which is a multi-k-linear map of degree 2 − d satisfying the relations
k
1
+k
2
=k+1
(−1)
∣x
1
∣+⋯+∣x
i−1
∣+i−1
m
k
1
(x
1
, . . . , mk
2
(x
i
, . . . , x
i+k−1
), ⋯, x
k
) = 0.
How do these start? The first few are m
1
○m
1
= 0 and m
1
(m
2
(x
0
, x
1
))+m
2
(m
1
(x
0
), x
1
)+
(−1)
∣x
0
∣+1
m
2
(x
0
, m
1
(x
1
)) = 0.
Then there is the unit, which is that m
k
(⋯, e
x
, ⋯, 0) is zero, and that m
2
(e, x) =
x = (−1)
∣x∣
m
2
(x, e).
Some remarks, in general this is not a category. If m
k
= 0 for k ≥ 3 this is a dg
category.
Let me finish my talk by saing why these come into play in the ∞ category
setting. We want to measure some A
∞
algebra from the algebro-geometric side.
To that purpose, we should enlarge the A
∞
category. We have structures in
algebraic geometry, ⊕, [1], ⊗, cone.
So what do we do? We think about the Yoneda embedding.
