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11122
bio website preschema.com
location Essen, Germany
age 23
visits member for 4 years, 10 months
seen 1 hour ago

Aug
3
revised A model structure on marked simplicial sets
add tag higher-category-theory
Aug
3
suggested suggested edit on A model structure on marked simplicial sets
Jul
26
comment Is Independent University of Moscow recognized?
It is not advisable (or probably even possible) to do your bachelor's studies in only IUM. Attending courses there is likely to increase your chances of being accepted in a North American or European graduate school (just speaking from personal experience), but you should do your degree somewhere else. If you want to do it in Russia (and speak Russian), HSE is a good option: math.hse.ru/en They have close ties to IUM and offer a well-respected bachelor's degree (with many students attending "special topics" courses in IUM as supplements).
Jul
24
comment When does prolongation preserve sheaves?
@user52824, I see the confusion: note that I am talking about presheaves on the site of all S-schemes. (Probably we should stop hijacking David's question now.)
Jul
24
comment When does prolongation preserve sheaves?
@user52824, $f^*$ preserves sheaves iff $f$ preserves covering families. When $f$ is the base change functor induced by a morphism of schemes, this is obviously true by the stability under base change of covering families.
Jul
24
comment When does prolongation preserve sheaves?
@DavidCarchedi, yes, I meant continuous in the sense of SGA 4.
Jul
24
comment When does prolongation preserve sheaves?
Just an obvious remark: in case $f : C \to D$ admits a right adjoint $g$, then $f_!$ preserves sheaves iff $g$ is continuous.
Jul
24
comment When does prolongation preserve sheaves?
@user52824, actually the base change functor $f : \mathrm{Sch}(T) \to \mathrm{Sch}(S)$ induced by a morphism of schemes $S \to T$ preserves covering families for any topology, and so $f^*$ always preserves sheaves. So this is really a reasonable condition (though at the same time I doubt it has anything to do with $f_!$ preserving sheaves).
Jul
19
revised Good properties of the $H^0$ functor (from quasi-functors to ordinary functors)
added 68 characters in body
Jul
19
answered Good properties of the $H^0$ functor (from quasi-functors to ordinary functors)
Jul
11
comment Is there any publication of “Beilinson’s dream” on motivic (complexes of) sheaves?
Have you looked at the book of Cisinski-Deglise (Triangulated categories of mixed motives)?
Jul
3
awarded  Informed
Jun
25
comment Is there a left-adjoint to the restriction of comodules?
I believe the answer is no. The corestriction of scalars functor $f_* : \mathrm{Comod}(C) \to \mathrm{Comod}(D)$ does preserve finite limits (they are just limits of the underlying vector spaces). However, infinite products of comodules are not just products of the underlying vector spaces so there is no reason for $f^*$ to preserve them.
Jun
22
comment Verdier localization for stable $\infty$-categories
Right. However I do not how to prove that $\mathrm{ho}(\mathcal{T}/\mathcal{C}) = \mathrm{ho}(\mathcal{T})/\mathrm{ho}(\mathcal{C})$, i.e. the homotopy category of the $\infty$-Verdier localization is the Verdier localization of the homotopy categories.
Jun
22
awarded  Nice Question
Jun
22
comment Verdier localization for stable $\infty$-categories
In the DG setting, the DG quotient constructions of Keller and Drinfel'd are models for the homotopy cofibre (in the Morita model structure on the category of small DG categories). This suggests defining the Verdier localization of a stable $\infty$-category by a stable sub-$\infty$-category as the cofibre of the inclusion. By (Higher algebra, Proposition 1.1.4.6) this is again a stable $\infty$-category.
Jun
12
comment Equivariant motivic sheaves
@EldenElmanto, that is a good review, but it does not discuss SH over diagrams of schemes.
Jun
10
comment Equivariant motivic sheaves
@ReladenineVakalwe, I should also have mentioned section 2.4.4, particularly the subsection titled "Les quatre operations pour les S-morphismes", where he shows that the results of chapter 1, about cross functors, extend to diagrams of schemes. I think the idea is that, given what he calls a stable algebraic homotopic derivator, one gets an induced "cross functor", which is basically the yoga you want (see section 1.4.1).
Jun
10
comment Equivariant motivic sheaves
@ReladenineVakalwe, you can try to take a look at the introduction of the thesis, and the introductions of the various chapters. In section 4.5 he defines the stable motivic homotopy category over a diagram of schemes (e.g. over a simplicial scheme). This gives a contravariant 2-functor from the category of diagrams of schemes to the category of symmetric monoidal triangulated categories (4.5.24). It does admit the "yoga of the six functors" in the sense that all the properties listed in Definition 2.4.12 are verified.
Jun
8
comment What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
@DanPetersen, SH embeds fully faithfully into SH(Spec(C)). See [Marc Levine, A comparison of motivic and classical stable homotopy theories, uni-due.de/~bm0032/publ/MotVClass.pdf].