2,144 reputation
11427
bio website preschema.com
location Essen, Germany
age 23
visits member for 5 years, 4 months
seen 7 hours ago

Apr
9
comment Algebraic K-theory of complex varieties
Analytic descent for K-theory would follow directly from analytic descent for perfect complexes. I have no idea whether the latter is true, though.
Apr
6
comment Integral transform on noncommutative spaces
@bananastack, I learned this from Marco Robalo's thesis, but it is probably in some paper of To\"en.
Apr
6
comment How does one compute a colimit of monoidal categories?
I believe non-strict monoidal categories can be described as algebras over a monad which is a cofibrant replacement of the monad presenting strict monoidal categories. So a similar statement should hold.
Apr
6
comment Integral transform on noncommutative spaces
@bananastack, saturated dg-categories also come from dg-algebras, by the way (from dg-algebras of finite type, even).
Apr
5
comment What is the applications of the dg-enhancements of derived categories of sheaves
@ZhenLin, I notice now that this is made precise, in terms of dg-localization, in section 2.4 of To\"en's notes here.
Apr
4
comment What is the applications of the dg-enhancements of derived categories of sheaves
@ZhenLin, this wasn't meant to be a precise statement, but one can probably use the dg-localization or also a spectral analogue instead to make it precise.
Apr
1
comment NCG with all noncommutativity in a nilpotent ideal
In line with Qiaochu's comment, some aspects of E_n-geometry are studied in the thesis of John Francis and this sequel.
Mar
30
comment When is the category of small (pre)sheaves a(n elementary) topos?
According to this paper of Mike Shulman, when $\mathcal{P}C$ is finitely complete, it is an infinitary pretopos, so by Giraud's theorem it is a topos iff it admits a small generating set.
Mar
27
comment Is the derived category of perfect complexes idempotent complete?
In the language of infinity-categories, the relevant result is Proposition 5.5.7.8 in Higher Topos Theory. See also Section 2.4 of arxiv.org/pdf/1001.2282.pdf, especially the text right before Lemma 2.20, for a discussion adapted to the stable setting.
Mar
27
comment Is the derived category of perfect complexes idempotent complete?
One good reference is Proposition 2.1.1 here: arxiv.org/abs/math/0204218.
Mar
25
comment Do algebraic stacks satisfy fpqc descent?
@Niels, for me an algebraic stack is defined to be a sheaf of groupoids (satisfying some conditions), so I interpret the question as whether or not this sheaf satisfies fpqc descent.
Mar
23
comment When do we have $D_{\text{perf}}(\text{Qcoh}(X))\simeq D_{\text{perf}}(X)$?
The functor you write is not an equivalence, but is rather fully faithful with essential image spanned by the complexes with quasi-coherent cohomology. That said, your claim is still true. I doubt that the assumptions can be relaxed, though. Verdier's counterexample in SGA 6 shows that this is unreasonable without something stronger than quasi-separated. While without quasi-compactness, it is not even true that perfect complexes are cohomologically bounded. What kind of generality were you hoping for?
Mar
13
comment Brandt's definition of groupoids (1926)
What could you mean by "categories are seen as special $\infty$-groupoids"?
Mar
12
comment Choice of fibrations is like a choice of a basis of a module
This is a well-known analogy which I first learned from Chris Schommer-Pries's answer here: mathoverflow.net/a/78408/2503
Mar
11
comment Can motivic E_∞-ring spectra be strictified to commutative motivic symmetric ring spectra?
Note: the proposition cited from Higher Algebra is now numbered 4.5.4.7 in the new version.
Mar
6
comment Explict form of $E_\infty$-morphisms between differential graded commutative algebras
Sorry, that was just a guess, and is probably wrong.
Mar
6
comment Explict form of $E_\infty$-morphisms between differential graded commutative algebras
According to Theorem 0.3 of arxiv.org/pdf/1412.1255v1.pdf, an $A_\infty$-functor between dg-categories is the same thing as a right quasi-representable bimodule. Since $A_\infty$-morphisms between commutative dg-algebras are the same thing as $E_\infty$-morphisms ($E_\infty$-alg $\hookrightarrow$ $A_\infty$-alg is fully faithful, right?), this seems to give an explicit description in terms of certain modules.
Mar
3
comment When does the canonical model structure on $\mathcal V$-$\mathbf{Cat}$ give a structure of monoidal model category?
dgCat is also not a monoidal model category (see mathoverflow.net/questions/195178/…).
Feb
25
comment hypothetical model structure on the category of model categories
@DmitriPavlov: the category of categories is, but I don't know about the category of model categories.
Feb
23
comment Does the stable category of a nice exact category embed in (the underlying category of) a derivator?
Is your question answered by the paper webusers.imj-prg.fr/~bernhard.keller/publ/…?