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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].
Jun
8
comment What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?
@user125763, SGA 4?
Jun
3
comment DG enhancements of $\ell$-adic derived categories
@ReladenineVakalwe, yes, one may define a dg-enhancement of the $\ell$-adic derived category as the homotopy limit of the dg-enhancements of certain "real" derived categories (which of course have dg-enhancements by abstract nonsense). This is the approach taken in the paper (Liu-Zheng, Enhanced adic formalism and perverse t-structures for higher Artin stacks). (There they use the language of $\infty$-categories, but this doesn't really matter of course.)
May
30
comment What information is lost in $X \to \mathrm{Sh}(X)$?
See (Mac Lane-Moerdijk, Sheaves in geometry and logic, Section IX.5, Proposition 2), for a proof of the fact that the functor taking a locale to its topos of sheaves is fully faithful. (Recall that the category of locales with enough points is equivalent to the category of sober topological spaces.)
May
25
awarded  Yearling
May
25
answered DG categories - pre-triangulated versus small limits
May
25
comment Morphism between Fourier-Mukai functors implies the morphism between kernels?
Don't you rather want to say something like the (triangulated) functor category Fun(D(X), D(Y)) should be replaced by the category of dg-functors?
May
15
comment References for the “nerve of an algebraic variety”
I would say that the choice of $f$ as $\Delta^\bullet$ is rather canonical, and really the only way you will get something that could be called a nerve. However you are right that, as Tom Goodwillie also pointed out, this doesn't give a very interesting simplicial set for $C = Sch$. But as I tried to explain in my answer, when $C$ is a ($\infty$-)cocompletion of $Sch$, there is an $sSet$-enrichment, and thus one gets the "richer" sSet-enriched version of the nerve which I denoted $\underline{Hom}(\Delta^n, -)$. And this is certainly something interesting.
May
15
revised References for the “nerve of an algebraic variety”
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May
15
comment References for the “nerve of an algebraic variety”
@QiaochuYuan, there is some confusion because what he is describing is actually the geometric realization $sSet \to C$, which is left adjoint to the nerve $C \to sSet$. The nerve is just a restriction of scalars which always exists, but one needs to assume $C$ is cocomplete to get a left adjoint. What you point out is not a problem because he does assume $C$ to be cocomplete in the answer, but even if it is not, one can extend to a cocomplete category (by the Yoneda embedding, for example). This will give a good notion of nerve and geometric realization $PSh(C) \rightleftarrows sSet$.
May
15
revised References for the “nerve of an algebraic variety”
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May
14
revised References for the “nerve of an algebraic variety”
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May
14
revised References for the “nerve of an algebraic variety”
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May
14
revised References for the “nerve of an algebraic variety”
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May
14
answered References for the “nerve of an algebraic variety”
May
13
comment Universal property of gluing [collage, cograph] of dg-categories
See Remark 4.2 in the linked paper.