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bio website preschema.com
location Essen, Germany
age 23
visits member for 5 years, 3 months
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7h
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.
13h
answered Monoidal structure on simplicial sheaves
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
25
awarded  Explainer
Mar
25
answered Do algebraic stacks satisfy fpqc descent?
Mar
25
revised Do algebraic stacks satisfy fpqc descent?
fix link, which wasn't working
Mar
25
suggested approved edit on Do algebraic stacks satisfy 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
4
revised Why are pushouts the right tool in these setups
added 8 characters in body
Mar
4
answered Why are pushouts the right tool in these setups
Mar
3
revised A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category
added 50 characters in body
Mar
3
answered A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category
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/…).