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11122
bio website preschema.com
location Essen, Germany
age 23
visits member for 4 years, 10 months
seen 5 hours ago

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”
added 14 characters in body
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.
May
13
comment Universal property of gluing [collage, cograph] of dg-categories
@FrancescoGenovese, they are the same for pretriangulated dg-categories, and otherwise the version of Orlov is the pretriangulated envelope of the version of Kuznetsov-Lunts.
May
4
comment Morphism between Fourier-Mukai functors implies the morphism between kernels?
However, the corresponding statement at the level of dg-categories is true: $\mathbf{D}(X \times Y) = \underline{\mathbf{Hom}}(\mathbf{D}(X), \mathbf{D}(Y))$, so that morphisms of integral kernels correspond to morphisms of the induced dg-functors (note though that the Hom means the internal Hom in the localization of the category of dg-categories with respect to the quasi-equivalences).
May
2
comment Comparison of model structures
@user38585, see section 1.3.3. In particular Corollary 1.3.16 is a useful way to check if a given Quillen adjunction is a Quillen equivalence.
Apr
17
comment Could one recover the relative K-theory from the quotient derived category?
Given dg enhancements of the derived categories, the Verdier quotient may be described as the homotopy category of the homotopy cofibre in the category of small dg categories with the Morita model structure. As a functor on the category of dg-categories, K-theory is known to preserve homotopy cofibres (this is part of being a localizing invariant in the sense of Tabuada).
Mar
6
awarded  Nice Answer