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2
votes
0answers
80 views

$B_\infty$ algebra structure on $C(\mathcal{A}, \mathcal{A})$

If $A$ is an associative unital dg algebra it is known ( see for example: http://arxiv.org/abs/hep-th/9403055) that its Hochschild complex has the structure of a $B_\infty$ algebra. Given a dg ...
2
votes
0answers
142 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) ...
8
votes
2answers
707 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
0answers
236 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
194 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
181 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
673 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
251 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
782 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
806 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 ...
3
votes
0answers
515 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 To├źn's notes, "Lectures on dg-categories" http://www.math.univ-toulouse.fr/~toen/swisk.pdf and in particular I would like to understand ...
5
votes
0answers
344 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
775 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 ...
6
votes
0answers
924 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 ...