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Oct
14
revised Is dgCat a category or a 2-category?
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Oct
14
comment Is dgCat a category or a 2-category?
Any model category (in fact, even just a category with weak equivalences) gives rise to an $(\infty,1)$-category. For example, if one takes simplicially enriched categories as a model of $(\infty,1)$-categories, this is given by the Dwyer-Kan simplicial localization.
Sep
29
comment Is dgCat a category or a 2-category?
@ZhenLin: Gepner-Haugseng show that the $\infty$-category of $V$-enriched $\infty$-categories has an enrichment over itself, for $V$ a presentably $E_2$-monoidal $\infty$-category. In particular this gives the enrichment of DGCat over itself, and hence via Dold-Kan an enrichment over $\infty$-categories. (I'm using implicitly the rectification result of Haugseng, which implies that DGCat is equivalent to the $\infty$-category of $\infty$-categories enriched over the $\infty$-category of chain complexes.)
Sep
29
answered Is dgCat a category or a 2-category?
Sep
27
awarded  Nice Answer
Sep
26
comment $E_{\infty}$ $R$-algebras vs commutative DG $R$-algebras vs simplicial commutative $R$-algebras
Yes, the functor $\phi$ just sends a simplicial algebra to its "normalized cochain complex".
Sep
26
answered Derived algebraic geometry: how to reach research level math?
Sep
23
comment Why do the model structures on dg-algebras and on dg-categories are not compatible?
A small remark: the functor of $\infty$-categories from (dg-algebras) to (dg-categories) factors through a functor to the $\infty$-category of pointed dg-categories which is fully faithful. As you point out, the mapping spaces in (pointed dg-categories) are obtained from the mapping spaces in (dg-categories) by quotienting out by the action of the simplicial group of units.
Sep
16
awarded  Nice Answer
Jun
26
awarded  Notable Question
Jun
18
comment Higher refinement of Seifert-van Kampen theorem on the language of hocolim
@DavidRoberts, Lurie's version is discussed here: ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem
May
25
comment is there a moduli of stable infinity categories?
@pro, thanks. I know that moduli stacks of dg-categories are discussed in To\"en's beautiful paper on derived Azumaya algebras, which has been generalized to a spectral stack parametrizing R-linear categories for R a commutative ring spectrum by Antieau-Gepner. I don't think they discuss the type of issue you are interested in there, though.
May
24
comment Homotopy pullback preserving functor
@FernandoMuro, I may be missing something, but if $A$ and $B$ have the trivial model structures, then the condition of preserving fibrations and weak equivalences is vacuous, while the condition of preserving homotopy fibres is a type of left-exactness, right?
May
24
comment Homotopy pullback preserving functor
@FernandoMuro, the proposition in the paper says that the functor preserves fibrations in the model structure, not fibre sequences.
May
24
comment is there a moduli of stable infinity categories?
@pro, thanks! Would you mind also sharing the approximations that you found in the literature?
May
24
comment is there a moduli of stable infinity categories?
@pro, would you mind explaining what you mean by infinitesimal theory here?
May
14
revised Matrix factorizations as a dg-category?
edited tags
May
13
answered How to show the following two definitions of homotopy monomorphism are equivalent?
May
13
answered A question about the morphisms in the homotopy category of dg-Cat
May
3
awarded  Custodian