CGP DERIVED SEMINAR
GABRIEL C. DRUMMOND-COLE
1. Youngjin Bae
Let me briefly give you what I want to do in one hour. We are considering the
model categories and dg categories, and we want to consider M a model category
enriched by chain complexes over k, and this is a model category and also has a dg
category structure, and I want to say, if I consider the dg category, I can consider
Int(M), and I can consider the model category, let me say M . I can consider the
homotopy category of the dg-category HoInt(M). We can also consider the derived
category of this model category W
−1
M, and I want to compare these. At the last
point I want to define a kind of Yoneda embedding in the case of dg categories.
That’s my brief outline and introduction.
Definition 1.1. Let me recall the C(k)-model category structure. This consists
of the data of:
(1) A C(k)-module structure on M, ⊗ ∶ C(k) ⊗ M → M satisfying the associa-
tivity and unit conditions, E ⊗ (E
′
⊗ X) ≅ (E ⊗ E
′
) ⊗ X and k ⊗ X ≅ X for
all E and E
′
in C(k) and X in M.
(2) For any pair X and Y in my model structure, I have Hom(X, Y ) ∈ C(k) for
all X and Y in M satisfying that Hom(E, Hom(X, Y )) ≅ Hom(E ⊗ X, Y )
(3) M already has a model structure so we want a compatibility of the model
and the module structure. i ∶ E → E
′
is a cofibratino in C(k) and j ∶ A → B
is a cofibration in M , then we want
E ⊗ B ⊔
E⊗A
E
′
⊗ A → E
′
⊗ B
to be a cofibration in M
Definition 1.2. Let M be a C(k)-model category and T a dg-category. Then M
T
is called the T -dg module with coefficients in M , for objects dg functors from T
to M , which is the same thing on objects as F
x
∈ M for x ∈ T . This satisfies the
following compatibility: F
x
⊗ T (x, y) → F
y
with
F
x
⊗ T (x, y) ⊗ T (y, z)
//
F
x
⊗ T (x, z)
F
y
⊗ T (y, z)
//
F
z
and the unit condition F
x
⊗ 1
x
∼
Ð→ F
x
.
1
2 GABRIEL C. DRUMMOND-COLE
The morphisms from F → F
′
are collections f
x
∶ F
x
→ F
′
x
for x ∈ T with the
compatibility condition
F
x
⊗ T (x, y)
//
f
x
⊗1
F
y
f
y
F
′
x
⊗ T (x, y)
//
F
′
y
So these are just natural transformations.
Remark 1.1. M
T
admits a model category structure. I considered some dg cate-
gory but it has a model category structure. f is a weak equivalence in M
T
if f
x
is
a weak equivalence in M for all x. Also f is a fibration in M
T
if f
x
is a fibration in
M. By the lifting property, the cofibrations of M
T
are
⧄
(W
M
T
∩ Fib
M
T
). This de-
fines a model category structure (under some conditions, i.e., that M is cofibrantly
generated)
Remark 1.2. M
T
admits a C(k)-model structure
Some exercise. Let T and T
′
be dg categories and M a C(k)-model category.
Then I want to give an idea of the proof that M
T ⊗T
′
≅ (M
T
)
T
′
as C(k)-model
categories.
In order to make this make sense, we need M
(T ⊗T
′
)
to be a dg model category.
Proof. We’re going to define a map Φ from M
(T ⊗T
′
)
→ (M
T
)
T
′
. So on objects,
F ↦ Φ(F ).
So F ∶ T ⊗ T
′
→ M is a dg functor, with F
(x,x
′
)
such that F
(x,x
′
)
⊗ (T ⊗
T
′
)((x, x
′
), (y, y
′
)) → F
(y ,y
′
)
with associative and unit conditions.
We want to define Φ(F ) ∶ T
′
→ M
T
. This Φ(F )
x
′
for x
′
in T
′
with
Φ(F )
x
′
⊗ T
′
(x
′
, y
′
) → Φ(F )
′
y
with Φ(F )
x
′
a functor T → M, which is a dg functor with the following data
(Φ(F )
x
′
)
x
for x in T , such that there are maps
(Φ(F )
x
′
)
x
⊗ T (x, y) → (Φ(F )
x
′
)
y
for x and y in T . The answer is then quite obvious, we choose Φ(F ) satisfying
(Φ(F )
x
′
)
x
= F
x,x
′
in M . You can directly check that using the morphism (T ⊗
T
′
)((x, x
′
), (y, y
′
)) ≅ T (x, y) ⊗ T
′
(x
′
, y
′
) and this gives your compatibility, you can
cook up the things you need.
The morphisms are also roughly the same.
Φ ∶ M
(T ⊗T
′
)
(F, F
′
) → (M
T
)
T
′
(Φ(F ), Φ(F
′
))
and maybe this is boring and I’ll skip it.
Okay, so now for M a C(k)-model category, we have M
a dg-category with
objects the objects of M and morphisms M(x, y) = Hom(x, y) from the C(k)-
enrichment. But we actually want Int(M), which is the full sub-dg-category whose
objects are M
cf
, the fibrant and cofibrant objects of M , that is, the ones where
∅ → X is a cofibration and Y → ∗.
If I consider [Int M ], I can also consider Ho(M)
∶
= M
cf
/ ∼ and I want to say
something about the ∼ homotopy relation for M
cf
.
DERIVED SEMINAR 3
So I have maps f and g from x to y, and I want a cylinder x ⊔ x
i⊔j
ÐÐ→ C(x)
p
Ð→ x
where i ⊔ j is a cofibration and p is a fibration and p ○ i = p ○ j is the identity on x,
and then we want
x
i
f
!!
C(x)
h
//
y
x
j
OO
g
==
This is “left” homotopy
There is also “right” homotopy, using a path object, a factorization of Y
s
Ð→
P (Y )
p×q
ÐÐ→ Y × Y which factorizes the diagonal into a trivial cofibration followed by
a fibration, and then you want
y
x
f
==
g
!!
h
//
P (y)
p
OO
q
y
So at the object level both [Int M] and M
cf
/ ∼ are the same. The morphisms of
[Int M ] are H
0
(IntM(x, y)) where Int is the same as Hom on our full subcategory
and so we want to show
H
0
(Hom) ≅ Hom
M
(x, y)/ ∼
So this was very hard for me, but the fact you use is that
Hom
M
(x, y) ≅ Hom
M
(k ⊗ x, y) ≅ Hom
C(k)
(k, Hom(x, y))
so the right hand side are degree zero chain maps in Hom(x, y). So these are
Z
0
(Hom(x, y)).
So the left hand side is Z
0
(Hom(x, y))/B
0
(Hom(x, y)), and so the only thing we
need to do is to compare the model category homotopy relation with the quotient
by B
0
(Hom(x, y)).
So on the left we say f − g = dh, and then I have the following, suppose I have a
right homotopy, Hom(x, y)Hom(x, P (y)) ⇉ Hom(x, y), and I want to say that this
uses f − g = dh
′
[Some discussion about how Hom(x, P (y)) ≅ P (Hom(x, y)) and
this can be used to reduce to chain complexes.]
