Questions tagged [dg-categories]
A differential graded category is a category enriched over complexes of modules for some commutative ring.
8
questions
9
votes
2
answers
636
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 $D$...
6
votes
2
answers
1k
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 $\mathcal V =\mathbf C(k)$ ...
30
votes
3
answers
4k
views
DG categories in algebraic geometry - guide to the literature?
Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric ...
21
votes
2
answers
3k
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?).
...
10
votes
1
answer
694
views
Why do the model structures on dg-algebras and on dg-categories are not compatible?
First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write $CdgAlg\...
8
votes
0
answers
425
views
Relationship between different definitions of the Hochschild homology
Throughout the literature, one can find many definitions of the Hochschild homology of various objects. However, the precise relationship between these definitions is not always so clear, at least to ...
6
votes
1
answer
256
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 $\...
3
votes
1
answer
232
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 dg-...