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Results tagged with higher-category-theory
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user 2503
For questions involving one or more categorical dimensions, or involving homotopy coherent categorical structures.
14
votes
What is the relationship between connective and nonconnective derived algebraic geometry?
As Tyler pointed out, it is "too easy" to be representable in the non-connective world. This might sound good, but it comes at the cost of geometric intuition. It is related to the fact that negativ …
- 5,901
6
votes
Accepted
Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?
Toen proved that $RHom$ provides the internal hom in the homotopy category of dg-categories. For what you want to be true, you need something more than this: you need to know that $RHom$ is actually …
- 5,901
14
votes
Accepted
Geometric morphism of $\infty$ topos
I never found any discussion about this in HTT either, but it turns out to work exactly the same way as in SGA 4.
That is, let $f^* : P(D) \to P(C)$ denote the restriction functor on presheaves, and …
- 5,901
10
votes
Accepted
Is dgCat a category or a 2-category?
The model structure on the category of dg-categories presents an $(\infty,1)$-category DGCat. This structure is essentially provided by the existence of mapping spaces (or mapping $\infty$-groupoids) …
- 5,901
5
votes
Accepted
A question about the morphisms in the homotopy category of dg-Cat
More generally one has the following statement: if $u : C \to D$ is a quasi-fully faithful functor of dg-categories, then the induced morphism of mapping spaces in the model category of dg-categories …
- 5,901
2
votes
Accepted
Integral transform on noncommutative spaces
Let $DGCat_k$ denote the $\infty$-category obtained by localizing the category of dg-categories at Morita equivalences. This is presented by the Morita model structure on the category of dg-categorie …
- 5,901
15
votes
Accepted
What is the applications of the dg-enhancements of derived categories of sheaves
It is hard to know where to begin! A general principle is that as long as you are only concerned with the derived category of a single variety, it is generally sufficient to consider it as a triangul …
- 5,901
14
votes
Accepted
Relationship between Hochschild cohomology and Drinfeld centers
Classically, Hochschild cohomology is an invariant defined for associative algebras while the Drinfeld centre is an invariant defined for monoidal categories. The latter is a categorification of the …
- 5,901
3
votes
Good properties of the $H^0$ functor (from quasi-functors to ordinary functors)
In the framework of $\infty$-categories, I think it is not difficult to see that your claim for $\mathcal{A} = \Delta^1$ holds for all $\infty$-groupoids.
Let $K$ be an $\infty$-groupoid (Kan complex …
- 5,901
3
votes
DG categories - pre-triangulated versus small limits
At least one direction of the question is answered by the paper
Giovanni Faonte, Simplicial nerve of an A-infinity category, arXiv:1312.2127.
See section 4.2, where he proves that for a pre-triang …
- 5,901
37
votes
3
answers
6k
views
Conjectures in Grothendieck's "Pursuing stacks"
I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to this questi …
- 5,901