A differential graded category is a category enriched over complexes of modules for some commutative ring.
11
votes
0answers
336 views
Lifting DG-categories to characteristic zero
The question of lifting (smooth projective) varieties from an algebraically closed field $k$ of characteristic $p$ to characteristic zero (i.e., to the Witt vectors $W(k)$) is a classical one. It's ...
6
votes
2answers
107 views
An example of two cofibrant dg categories whose tensor product is not cofibrant
I have been reading the paper by Toën "The homotopy theory of dg categories and derived Morita theory" where in chapter 4 it is stated that the tensor product of two cofibrant dg categories $C$ and ...
0
votes
0answers
59 views
Extending natural transformations of triangulated functors on $D^b(\mathbb P^1)$
Let $\mathcal T = D^b(\mathbb P^1)$, the bounded derived category of $\mathbb P^1$ over a field $k$. It is well known that $\mathcal T$ admits a strong and full exceptional sequence $\{\mathcal O, ...
0
votes
0answers
62 views
Complex of “homotopy coherent dg-natural transformations” between dg-functors
Let $\mathcal B$ be a dg-category. I define the dg-category of morphisms $\underline{\mathrm{Mor}}(\mathcal B)$ as follows: objects are triples $(A,B,f)$, where $A,B \in \mathcal B$ and $f \in ...
1
vote
1answer
90 views
Is the extension of a quasi-functor again a quasi-functor?
Let $\mathcal A, \mathcal A', \mathcal B$ be dg-categories over a field $k$ (this assumption allows me not to derive the tensor product, I don't think it is really essential). Let $F : \mathcal A \to ...
3
votes
0answers
126 views
Perf($\mathscr{A}$) and perfect chain complexes
Suppose $\mathscr{A}$ is a dg category and $ \mathcal{D} (\mathscr{A})$ its associated derived category. For an object in $ a \in \mathscr{A}$ there is associated (right )dg $ \mathscr{A}$ module, $ ...
3
votes
0answers
102 views
Lifting commutative diagrams of functors from the homotopy level to the “higher” level
Let $\mathcal A$ and $\mathcal B$ be differential graded categories over a field. Let $F, G, K : \mathcal A \to \mathcal B$ be quasi-functors (see here for definitions). Assume you have morphisms of ...
5
votes
2answers
331 views
A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category
I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea.
Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
1
vote
1answer
271 views
(Homotopy) limits and colimits in a dg-category
It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of ...
1
vote
1answer
103 views
Taking zeroth cohomology of a dg-module over a dg-category is “good” on representables, but probably “bad” on cones
This question is possibly related to this other one.
Let $\mathcal A$ be a dg-category over a commutative ring $k$. I denote by $\text{dgm-}\mathcal A$ the dg-category of right dg-$\mathcal ...
1
vote
0answers
44 views
Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones
This question is strictly related to this other one.
Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let
be a diagram in $Z^0(\mathcal A)$, where the rows are ...
3
votes
1answer
105 views
Reconstructing a morphism of exact triangles in the homotopy cat. of a dg-cat. using a “functorial cone map”
I set this problem in the framework of (pretriangulated) dg-categories; everything can probably be translated in the world of stable $(\infty,1)$-categories.
Let $\mathcal A$ be a pretriangulated ...
4
votes
1answer
112 views
Good properties of the $H^0$ functor (from quasi-functors to ordinary functors)
Let $\mathcal A, \mathcal B$ be dg-categories over a field $k$. I denote by $\mathcal{RHom}(\mathcal A,\mathcal B)$ the dg-category (defined up to quasi-equivalence) which gives the internal hom in ...
2
votes
2answers
137 views
Does semi-free behave well under totalization
Suppose I have a dg algebra $(A,d)$ and a chain complex $M^\bullet$ of semi-free $(A,d)$ modules. I am hoping it is true that $ Tot^\coprod (M^\bullet)$ is again a semi-free $(A,d)$ module. Is this ...
4
votes
2answers
198 views
Universal property of gluing [collage, cograph] of dg-categories
In some recent works, such as this one (3.2, page 15), a definition of "gluing of dg-categories along a dg-bimodule" is given. It is obviously the analogue of the notion of collage (or cograph) of a ...
3
votes
0answers
188 views
Quasi-equivalences of DG categories
There are several definitions of a quasi-equivalence $\newcommand{\T}{\mathscr{T}}F : \T \to \T'$ of DG categories in the literature, e.g.
(i) the induced functor $H^0(F) : H^0(\T) ...
9
votes
2answers
912 views
Stable infinity categories vs dg-categories
What is the relation between dg-categories and stable $\infty$-categories?
Given a dg-category one can form its dg-nerve and get a $\infty$-category
(which will be stable if the dg-category is?).
...
6
votes
1answer
310 views
[Reference Request] The Definition of Adjoint Functors between dg-categories
Let $A$ and $B$ be two dg-categories, $F: A \rightarrow B$ and $G: B \rightarrow A$ are two functors. Then what is the definition that $F$ and $G$ form an adjoint pair?
In my mind $F\dashv G$ ...
4
votes
0answers
202 views
Formality of $A_\infty$-category vs formality of its total algebra
Let $\cal C$ be an $A_\infty$-category and $A$ its total algebra (elements in $A$ are formal linear combinations of arbitrary morphisms in $\cal C$ and multiplications of arrows which can't be ...
1
vote
1answer
202 views
When do dg-lifts exist?
Let $\mathcal A$ and $\mathcal B$ be abelian categories (with enough injectives and countable products) and $$F:D^+(\cal A) \rightarrow D^+(\cal B)$$
be a triangulated functor. I am interested ...
9
votes
2answers
733 views
Does the Riemann-Hilbert Correspondence work at the DG level?
let $X$ denote a smooth complex algebraic variety. Let $D_{rh}(X)$ denote the category of regular holonomic $D$-modules on $X$ and $D_{rh}^b(D(X))$ denote the bounded derived category of $D$-modules ...
7
votes
0answers
262 views
DG vs. abelian quotients
The following, if true, should probably be "standard," but I don't know where to look. I'd rather be slightly imprecise about hypotheses in the hope that there's a good general answer. Feel free to ...
4
votes
1answer
831 views
homotopy limits of dg categories
The question is related to the following MO question
(Co-)Limits and fibrations of DG-Categories?
My question is,
how to define the homotopy limit (and colimit) of a system of dg-categories ...
8
votes
1answer
962 views
What is the Hochschild cohomology of the dg category of perfect complexes on a variety?
Let $X$ be a quasi-projective variety over a field $k$. Let $D_{qcoh}$ be a dg enhancement of the unbounded derived category of quasi-coherent sheaves over $X$, and $D_{perf}$ its full subcategory of ...
4
votes
0answers
560 views
A Question about a theorem in Toën's notes “Lectures on dg-categories”
So I am trying to learn a bit about dg categories from Toën's notes, "Lectures on dg-categories" http://www.math.univ-toulouse.fr/~toen/swisk.pdf and in particular I would like to understand ...
6
votes
0answers
369 views
(Co-)Limits and fibrations of DG-Categories?
First of all, let me see if i got the 1-categorical version right:
Let $\mathcal F:C\to Cat $ be a
(pseudo-) functor. The 2-colimit
$\mathrm{colim}_C\mathcal F$ is then
given by the grothendieck
...
7
votes
1answer
787 views
dg objects: Z-graded vs. Z/2Z-graded
I am wondering: Are there any general theorems or principles relating the theory of Z-graded dg objects and the theory of Z/2Z-graded dg objects? I am mainly interested in dg algebras, dg Lie ...
7
votes
0answers
999 views
Regarding the Gerstenhaber bracket on Hochschild cohomology and Morita equivalence
Associated to any $A_\infty$ $k$-algebra $A$ the Hochschild cochain complex $CH^*(A)$ has the structure of a dg-Lie algebra and a dg-algebra which are compatible enough that the cohomology is a ...
