1,347 reputation
11425
bio website preschema.com
location Essen, Germany
age 23
visits member for 5 years
seen Dec 18 at 12:40

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.
Sep
16
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment 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
comment (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
comment 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.