bio | website | preschema.com |
---|---|---|
location | Essen, Germany | |
age | 23 | |
visits | member for | 5 years |
seen | Dec 18 at 12:40 | |
stats | profile views | 1,180 |
Nov 27 |
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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 |
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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 |
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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 |
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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 |
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Etale topos as a classifyng topos ?
The question is answered in arxiv.org/pdf/0902.1130v2.pdf (though surely known much earlier). |
Nov 8 |
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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 |
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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 |
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Does “simplicial” commute with “Bousfield localization”?
Of course, this has nothing to do with $\Delta$ and works more generally for presheaf (∞-)categories. |
Nov 2 |
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Why higher category theory?
See the closely related question mathoverflow.net/questions/169187/…. |
Oct 29 |
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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. |
Sep 16 |
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Can the Grothendieck ring of varities over a field $k$ be defined for non separated schemes?
@Ari, the question is not about the Grothendieck group $K_0(X)$ of a scheme X, but about the Grothendieck ring $K_0(Sch_k)$ of all (separated, finite type) k-schemes. |
Sep 15 |
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Are these two “FUNCTORS” adjoint?
I guess this is what the OP has in mind: ncatlab.org/nlab/show/spectrum+of+an+abelian+category |
Sep 13 |
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What is a stable $(n,1)$-category?
For n=2 there are apparently several definitions in the literature, see the overview article arxiv.org/abs/0904.0078. |
Sep 7 |
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Algebraic K-theory and Homotopy Sheaves
@user125763, I added a reference which seems to discuss this (though I haven't had a chance to look at it myself yet). |
Sep 5 |
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A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category
@MauroPorta, Lurie's construction does work in great generality, but probably is not that useful for practical purposes because one cannot really get a handle on this "chunk" (I am referring to Lemma A.3.4.6). For this reason I think the question raised by Francesco of in which generality the construction of Toën using quasi-functors works, is interesting. |
Aug 25 |
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what is the stabilization of pointed sets?
@tetrapharmakon, in general, if $C$ is a cocomplete and complete 1-category then its nerve $N(C)$ is the infinity-category generated by the trivial model structure on $C$. |
Aug 14 |
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Are the pullback functors of adjoint functors also adjoint?
If I recall correctly you can find this statement in SGA 4, in the exposé about functoriality of presheaf categories. |
Aug 13 |
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When do limits and colimits of infinity-categories commute?
Under appropriate assumptions one has: Proposition 5.3.3.3: filtered colimits commute with finite limits. And Lemma 5.5.8.11: sifted colimits commute with finite products. |
Aug 7 |
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(Homotopy) limits and colimits in a dg-category
This discussion on Kan extensions in simplicially enriched categories should translate directly to the dg-setting: ncatlab.org/nlab/show/homotopy+Kan+extension#InKanCplCat One then gets (co)limits as a special case. Alternatively, "most" dg-categories are actually dg-enriched model categories, and one can use the notion of homotopy (co)limit there. |
Aug 5 |
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Reconstructing a morphism of exact triangles in the homotopy cat. of a dg-cat. using a “functorial cone map”
Sorry, I was being silly. I think tetrapharmakon is right actually, the space of maps $w$ as above should be contractible. |