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

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.
Aug
5
comment Reconstructing a morphism of exact triangles in the homotopy cat. of a dg-cat. using a “functorial cone map”
@tetrapharmakon, the problem is the other direction: given a map $w : C(f) \to C(f')$ as above, it is not necessarily induced by some $u$ and $v$.
Jul
26
comment Is Independent University of Moscow recognized?
It is not advisable (or probably even possible) to do your bachelor's studies in only IUM. Attending courses there is likely to increase your chances of being accepted in a North American or European graduate school (just speaking from personal experience), but you should do your degree somewhere else. If you want to do it in Russia (and speak Russian), HSE is a good option: math.hse.ru/en They have close ties to IUM and offer a well-respected bachelor's degree (with many students attending "special topics" courses in IUM as supplements).
Jul
24
comment When does prolongation preserve sheaves?
@user52824, I see the confusion: note that I am talking about presheaves on the site of all S-schemes. (Probably we should stop hijacking David's question now.)
Jul
24
comment When does prolongation preserve sheaves?
@user52824, $f^*$ preserves sheaves iff $f$ preserves covering families. When $f$ is the base change functor induced by a morphism of schemes, this is obviously true by the stability under base change of covering families.
Jul
24
comment When does prolongation preserve sheaves?
@DavidCarchedi, yes, I meant continuous in the sense of SGA 4.
Jul
24
comment When does prolongation preserve sheaves?
Just an obvious remark: in case $f : C \to D$ admits a right adjoint $g$, then $f_!$ preserves sheaves iff $g$ is continuous.
Jul
24
comment When does prolongation preserve sheaves?
@user52824, actually the base change functor $f : \mathrm{Sch}(T) \to \mathrm{Sch}(S)$ induced by a morphism of schemes $S \to T$ preserves covering families for any topology, and so $f^*$ always preserves sheaves. So this is really a reasonable condition (though at the same time I doubt it has anything to do with $f_!$ preserving sheaves).
Jul
11
comment Is there any publication of “Beilinson’s dream” on motivic (complexes of) sheaves?
Have you looked at the book of Cisinski-Deglise (Triangulated categories of mixed motives)?
Jun
25
comment Is there a left-adjoint to the restriction of comodules?
I believe the answer is no. The corestriction of scalars functor $f_* : \mathrm{Comod}(C) \to \mathrm{Comod}(D)$ does preserve finite limits (they are just limits of the underlying vector spaces). However, infinite products of comodules are not just products of the underlying vector spaces so there is no reason for $f^*$ to preserve them.
Jun
22
comment Verdier localization for stable $\infty$-categories
Right. However I do not how to prove that $\mathrm{ho}(\mathcal{T}/\mathcal{C}) = \mathrm{ho}(\mathcal{T})/\mathrm{ho}(\mathcal{C})$, i.e. the homotopy category of the $\infty$-Verdier localization is the Verdier localization of the homotopy categories.