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bio website preschema.com
location Essen, Germany
age 23
visits member for 5 years, 1 month
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1d
comment When is the cofibrant replacement of a product the product of the cofibrant replacements?
This is true for coconnective dg-algebras, according to arxiv.org/abs/1112.2360.
Jan
23
comment Connection between quasifibrations and homotopy cartesian squares
Are you looking for something like this fibre-wise characterization: ncatlab.org/nlab/show/…?
Jan
17
comment Motivation for cyclic (co)homology
One motivation for cyclic homology and its variants is the utility in computing algebraic K-theory. See for example the Goodwillie theorem, [T. Goodwillie, Relative algebraic K-theory and cyclic homology, Ann. Math. 124 (1986), 347–402].
Jan
16
comment Homological algebra is linearized homotopical algebra?
Homological algebra is the homotopy theory of chain complexes. The homotopy theory of chain complexes is equivalent to the homotopy theory of modules over the Eilenberg-Mac Lane spectrum $H\mathbf{Z}$. Hence homological algebra is a stable, linear version of the homotopy theory of spaces. See [Stefan Schwede, Brooke Shipley, Stable model categories are categories of modules, Topology 42 (2003), 103-153].
Jan
16
comment Relationship between Hochschild cohomology and Drinfeld centers
@SamuelM, according to Remark 1.5 in the paper, it seems that the Drinfeld centre of $D(A \otimes A^{op})$ will be the $\infty$-category of modules over the Hochschild chain complex of $A \otimes A^{op}$.
Jan
14
comment Relationship between Hochschild cohomology and Drinfeld centers
@SamuelM, I think I finally understood what you were really asking. I updated my answer again, let me know if that helps.
Jan
13
comment Relationship between Hochschild cohomology and Drinfeld centers
@SamuelM, sorry, I wasn't very clear. I hope the updated answer is more helpful.
Jan
12
comment Which properties of a variety are detected by its derived category of coherent sheaves?
@მამუკაჯიბლაძე, when taking into account the derived tensor product, one can recover the variety completely (this is a theorem of Thomason and Balmer).
Jan
12
comment Which properties of a variety are detected by its derived category of coherent sheaves?
As far as I know, it is not possible to recover the derived category of quasi-coherent complexes from the bounded derived category of coherent sheaves, at the triangulated level. At the level of infinity- or dg-categories, one can recover it as the ind-objects (in the regular case).
Jan
12
comment Which properties of a variety are detected by its derived category of coherent sheaves?
The derived category detects homological invariants like (higher) algebraic K-theory, Hochschild homology, cyclic homology, etc. As for cohomological invariants, it is a theorem of Orlov that these are detected up to Tate twists in general, and in some special cases detected completely.
Nov
27
comment Site dependance of the Cech weak equivalences on simplicial sheaves
Just to be clear: you are asking if the weak equivalences in the model structure(s) described at ncatlab.org/nlab/show/… can be described independently of the site of definition, right?
Nov
17
comment Use of derivators to the theory of motives?
However, a beautiful idea of Ayoub is to modify the definition of derivator, replacing the 2-category of diagrams by the 2-category of diagrams of schemes over some base, and imposing various axioms. The resulting notion, which he calls an algebraic derivator, essentially captures the whole formalism of six functors. I am not sure, but I believe that Grothendieck probably did not have anything like this in mind when he wrote Pursuing Stacks or Les Derivateurs (and I will be very happy if an expert could confirm).
Nov
17
comment Use of derivators to the theory of motives?
In the motivic world, model categories and infinity-categories are used as refinements of triangulated categories. These are "stronger" refinements than derivators (in some precise sense), so I doubt that the theory of derivators can offer any improvements to the theory of motives.
Nov
11
comment A bestiary of topologies on Sch
Hi Pieter, I guess I should have waited for you to finish it and post it here yourself then! Still, I find it an invaluable resource already, so thanks a lot :)
Nov
10
comment Etale topos as a classifyng topos ?
The question is answered in arxiv.org/pdf/0902.1130v2.pdf (though surely known much earlier).
Nov
8
comment Mysterious quotes (at least for me)
Right, I should have said derived Morita invariance. And Orlov's remark was about the derived category of a commutative scheme.
Nov
6
comment why are motives more serious than “naive” motives?
@VivekShende, Chow motives are supposed to be the semisimple objects in the abelian category of mixed motives (whose derived category is supposed to coincide with Voevodsky's DM). There is a functor from Chow motives to DM which is known to be fully faithful.
Nov
3
comment Does “simplicial” commute with “Bousfield localization”?
Of course, this has nothing to do with $\Delta$ and works more generally for presheaf (∞-)categories.
Nov
2
comment Why higher category theory?
See the closely related question mathoverflow.net/questions/169187/….
Oct
29
comment Why care about Fourier-Mukai partners?
By results of D. Orlov and B. To\"en, there is no difference between equivalence at the level of derived categories or at the level of dg- or infinity-categories, for smooth projective varieties.