CGP DERIVED SEMINAR
GABRIEL C. DRUMMOND-COLE
1. June 12: Yong-Geun Oh
What I’m going to talk about is, try to give the proof of existence, of the model
category of dg categories. So let me denote DCAT as the category of differen-
tial graded categories. Somehow I read another paper of Tabuada, he introduces
another category DCAT
p
, which is the category of dg categories with one initial
category, with O, with morphisms of O → C in DCAT.
Theorem 1.1. (Tabuada) There is a model category structure whose weak equiva-
lences W are quasi-equivalences of dg categories. The generating set of cofibrations
is I, I’ll write it down in detail later. J is the generating set of trivial cofibrations.
So (W, J
⧄
,
⧄
(I
⧄
)) describe a model structure in DCAT
p
.
Let me describe a few categories. Let’s consider A a category with one object
called 3, with an identity endomorphism 3 → 3. Then B has two objects 4 and 5 and
each has just the identity morphism, and the morphisms between 4 and 5 are trivial.
Then there is a category K, with two objects 1 and 2, and a more complicated set
of morphisms. It has morphisms f ∶ 1 → 2 and g ∶ 2 → 1 and r
12
∶ 1 → 2. So in this
category Mor(1, 2) ≅ Hom
0
k
(1, 2) ≅ k, the ground ring, generated by f . Similarly,
Mor(2, 1) ≅ k, spanned by g. These are closed, that is, df = dg = 0. dr
1
= gf −1, with
r
1
∈ Hom
−1
k
(1, 1) and r
2
∈ Hom
−1
k
(2, 2), with dr
2
= f g − 1. Then dr
12
= f r
1
− r2
f
. So
what this means is, this is a kind of contraction, f intertwines these two morphisms.
So the morphisms in this category are generated by f , g, r
1
, r
2
, and r
12
.
This will be one slick way of expressing some condition.
So K and then there’s more. So here’s another category P(n), with objects 6
and 7, and the morphism structure has a morphism space D
n
from 6 to 7, this is
a complex, so what is D
n
? We denote by S
n−1
, this is a complex defined by
S
n−1
[i] =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
k i = n − 1
0 else
The analog of D
n
will have k in degree n − 1 and n, with the differential the
identity. Now let me introduce one more, R(n), which is the category of dg functors
from B to P(n). The objects are the same, but you have, well, R(n) is dg functors
from B to P(n) that send 4 to 6 and 5 to 7.
Here is another, C(n) is the dg category with two objects, 8 and 9, that has
morphisms S
n−1
from 8 to 9 and all other spaces minimal (k or 0 as appropriate).
Now I want to denote again by S(n) the unique dg functor from C(n) to P(n)
the dg functor that takes 8 to 6 and 9 to 7 and takes S
n−1
to itself in D
n
[long discussion about whether the differential goes down or up]
So Q is the dg functor from O to A. Now I’m going to tell you J.
1
