D G - CAT E G O R I E S I V
by g a b r i e l d r u m m o n d - c o l e
transcribed by da mi en l ej ay
My topic is the proposition mentionned by Tae-Su: let
T
be a
dg-category and let
M
be a
C(k)
-model category such that
(1)
,
M
is
cofibrantly generated and
(2)
(the thing with tensor produ). Then
[T, Int(M)] is equivalent in Set with Iso(HoM
T
).
[T, Int(M)] ' Iso(HoM
T
).
You would like to do things with only dg-category
M
but unfortunatly
you need the homotopy tools contained in model categories. This
theorem is a theorem that links the dg-side with the model category
side.
On the left you have a morphism set and on the right you have a
set of objes.
Let me make a comparison with the following context: if
R
is a ring
and V a veor space, then
Hom(R, GL(V) ) ' {R module rutures on V}
It is not true that every dg-category can be written as
Int(M)
for
M
a model category. But we can always use the Yoneda embedding
described by Tae-Su.
h : T
0
−→ Int(T
0op
-Mod)
And the proposition applies to
Int(T
0op
-Mod)
. And now I can apply
the proposition to [T, T
0
] inside
[T, Int(T
0op
-Mod)].
We get that
[T, T
0
]
can be identified with ‘those modules’ whose image
lies in the quasi-essential image of h.
Let us see an example: take
T
be the dg-category with only one
obje and Hom
T
(∗, ∗) ' k. Then
[k, T
0
] ⊂ Iso Ho(ob of T
0op
-mod)
And those would be the isomorphism classes of objes in
T
0
i.e objes
of [T
0
].
The Derived Seminar, June , Pohang, Korea.
©
Damien Lejay. All rights
reserved.
Other example, if
T
is the category with two objes
1, 2
and
Hom
T
(1, 2) ' k, then
[T, T
0
] ' morphisms in [T
0
].
If T is the ring R, then
[R, Int(C(k))] ' D(R).
Since I have only a few minutes left I am going to give some details
about the proof of the theorem.
Let me recall a lemma given by Tae-Su:
If
f : T
0
→ T
is a quasi-equivalence of dg-categories and
M
satisfies
(1) (2)
, then
f
!
and
f
∗
are Quillen equivalences between
M
T
and
M
T
0
.
Since we know that there exis a model ruure on the category
of dg-categories, there exis a cofibrant replacement
Q(T) ' T
then by the above lemma we get
M
Q(T)
' M
T
.
hence we can suppose that
T
is a cofibrant dg-category. A andard
thing in model category theory tells you that is you want to compute
the right hom set of morphisms you should choose a cofibrant domain
and a fibrant codomain. Here we may suppose that
T
is cofibrant and
it is true that Int(M) is a fibrant dg-category.
Surjeivity: any obje of
Ho(M
T
)
can be represented by a cofibrant
fibrant obje of M
T
. Any such funor takes values inside Int(M).
Injeivity: I wanna art with two maps
F, G
sitting in
Hom(T, Int(M))
such that
[i ◦ F] ' [i ◦ G]
and I wanna argue that they are equivalent.
For this, we shall use a path obje
Int(M)
T Paths
Int(M)
F
G
I am ju going to say that the path obje should look like the category
of morphisms of Int(M).
