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Regarding that the nlabizens have discussed so much various generalizations of Grothendieck topology, maybe somebody knows which terminology is convenient for the setup of covers of abelian categories by finite conservative families of flat localizations functors, or more generally by finite conservative families of flat (additive) functors. Namely the localizations functors do not mutually commute so the descent data are more complicated but if you produce the comonad from a cover then the descent data are nothing but the comodules over the comonad on the product of the categories which cover. In noncommutative geometry we often deal with stacks in this generalization of topology and use ad hoc language, say for cocycles, but the thing is essentially very simple and the language barier should be overcome. There are more general and ore elaborate theories of nc stacks, but this picture is the simplest possible.
I mean the main axiom which fails is the stabilization axiom. Rosenberg calls the case with this failure quasitopologies. Van Oystaeyen repalces this failure by complicated system of additional axioms which are interesting but not necessary for stack theory and the theory of noncommutative (I did not say only nonabelian!) cocycles.
Stabilization meaning stability under pullback? I don’t know of any term for a notion of cover which fails that; as Johnstone points out in the Elephant, it’s that condition which is indispensable to ensure that the sheaves you get for a notion of covering act like sheaves.
Right, stability under pullbacks. Quasicoherent sheaves in noncommutative algebraic geometry are well known among localization specialists to lack this property generically, but the descent/sheaf condition instead works for agreement on both consecutive compositions of localizations what replaces the categorical pullback intuitively.
For example take a ring $R$, a conservative family of Ore localizations $Q^*_i : R-Mod \to B_i$ where $B_i = S^{-1}_i R-Mod$ with right adjoints $Q_{i *} : B_i \to R-Mod$. Let $M$ be in $A = R-Mod$. Then the colimit of the diagram in $A$ consisting of $M$, all single localizations $Q_{i*}Q_i^*(M)$ and all consecutive localizations $Q_{j*} Q_j^* Q_{i*}Q_i^*(M)$ with all the natural adjunction etc. arrows in between is $M$. This is called the globalization lemma (proved in 1988 by Rosenberg by complicated methods but in fact it is just an easy corrollary of Beck’s theorem).
Of course to talk about noncommutative pretopologies we need to consider inverse image functors as geometric morphisms i.e. in opposite direction.
it’s that condition which is indispensable to ensure that the sheaves you get for a notion of covering act like sheaves
Sheaves are for passge from local to global, i.e. for descent. So I do not see why would somebody claim that pullback is the only notion generalizing intersection in categorical situations. Of course, the sheaves in the sense I am talking do not form a Gorthendieck topos; but are good for gluing, have natural stack generalizations, and what they do form is not that arbitrary.
Hi Zoran,
you have told me about this several times, of course, but each time I feel like I need to go back and remind myself of some details. Do we have an nLab page summarizin the key ingredients here?
Yes in my personal area there is an old article
gluing categories from localizations (zoranskoda)
in my perosnal part (I was quoting this before). One could also recommend section 2 (pages 8-13 esp.) of Kontsevich-Rosenberg 1998 Noncommutative smooth spaces where not localizations but noncommutative flat covers are used in a different formalism.
By “act like sheaves” I meant “form a topos.” It feels to me that through long usage, the word “sheaf” has come to imply certain behavior like exactness properties, such as being a topos, and if one wants to study gluing conditions that don’t satisfy those sorts of exactness one should call it something else, like “quasisheaf” or something.
It is still exact with respect to different notion of a cover. It indeed has a different name: noncommutative sheaf or sheaf over a noncommutative topology. Infinity topos is also not a topos, it is a generalization to infinity categorical situation. Noncommutative sheaf is a generalization to a situation which typically but not only appears in the study of local properties of spectra of noncommutative rings and generalizations. I do not defend the terminology, but this is important generalization and it deserves proper treatment, rather than saying it is not a topos. This generalization I talk here is appropriate to study quasicoherent noncommutative sheaves possibly with addiotional structure. On the other hand for sheaves of sets the other formalism generalizing Grothendieck topologies – the formalism of Q-categories can be used.
The KR article mentioned in 8 has a fantastic thing. Given a monoidal category C with some properties (I guess with equalizers commuting with tensor products) one considers the pairs (B,M) where B is a monoid in C and M an internal B-coring. They define morphisms and refinements, and localize at the class of refinements to get some category $Covers[Ref^{-1}]$. Then for a pair (B,M) they define the category of quasicoh sheaves; the morphisms induce of course direct and inverse image functors for qcoh sheaves. Now, they refine slightly the category Covers by adding additional data from the beginning. To the pair (B,M) they add certain structure map which is a epimorphism of B-corings $B\otimes B\to M$ (the meaning of this is interesting to ponder, but I will not write about it here). Then one defines morphisms by requiring also compatibility with structure morphisms and does a more subtle definition of refinement in this situation. The localization at refinements defines new category $NcSpaces = CoversSpaces[Ref^{-1}]$.
Now the theorem in the case when the monoidal category is the category of vector spaces over a field k: the category of quasicompact speareted schemes is equiv to a full subcategory of $CoversSpaces[Ref^{-1}]$. The category opposite to the category of associative algebras is also equivalent to a full subcategory of $CoversSpaces[Ref^{-1}]$. Thus the commutative algebraic geometry and noncommutative affine geometry are embedded by this simple device! Moreover the functor (B,M) mapsto Qcoh(B,M) extends to NcSpaces with usual results for commutative schemes and noncommutative affine schemes!
Also one can skip the condition that the structure epi is a map to get a convenient category essentially containing Artin stacks.
I’m happy with “noncommutative sheaf”. I wasn’t saying one shouldn’t study it, just that the reader should be warned that it’s more general than the more common notion of sheaf. If it is called a “noncommutative topology” then maybe that answers your original question?
Unfortunately, it does not. There are several different notions called noncommutative topology, and usually in nocommutative setup. However the notion of localization covers and of flat covers of that kind is much more general than the situation in noncommutative geometry. I mean localizations having fully faithful right adjoint happen everwhere, and there are many situations where some class of conservative families of localization induces a comonadic descent for some fibered category.
“noncommutative coverage”? (-: I don’t know.
I agree with MIke #14. The comment
I mean the main axiom which fails is the stabilization axiom.
by Zoran at #2 and the reply in #3
Stabilization meaning stability under pullback? I don’t know of any term for a notion of cover which fails that; as Johnstone points out in the Elephant, it’s that condition which is indispensable to ensure that the sheaves you get for a notion of covering act like sheaves.
by Mike leads me to say: what about coverages? They don’t have stability under pullbacks. E.g. good open covers on paracompact manifolds do not pull back to good open covers, but after pulling back they can be refined to a good open cover. In the Elephant these are used to define sheaves. Is there an analogous thing for the noncommutative case?
Also, for the generalised concept of topos, what about noncommutative topos? Sounds cool :)
Zoran wrote at #6
what they do form is not that arbitrary.
what sort of exactness properties do the categories of ’NC sheaves’ have? Are they cocomplete? finitely complete? etc…
In the notation of gluing categories from localizations (zoranskoda), is there a functor $Q_\mu A \to Q_\mu Q_\nu A$ to go along with $Q_\nu A \to Q_\mu Q_\nu A$? Does this make the square with top left corner $A$ and bottom right corner $Q_\mu Q_\nu A$ commute? If so then I think you might have a coverage (or cocoverage, if you want the arrows to point the way I’ve written here)
David, by “stability under pullback” I meant to include to the axiom satisfied by coverages, which is not literally a pullback in the category in question, but still says morally that given a map from U to V, any cover of V can be “pulled back” to a cover of U (though not necessarily in a universal way). That’s the essential axiom guaranteeing that sheaves form a topos.
@Mike
OK. In that case, I still feel that Zoran was referring to actual pullback, and probably thought you were too. Really the ’stability under pullback’ is, at least for singleton coverages, something like a cocartesian lift to a (co?)fibred category of arrows (a subcat of the usual category of arrows) over the base. I wonder how far that analogy can be pushed? It seems like something someone would have done already.
Zoran said
the sheaves in the sense I am talking do not form a Grothendieck topos
whereas the sheaves for a coverage, with its non-actual-pullback-stability condition, do form a Grothendieck topos.
something like a cocartesian lift to a (co?)fibred category of arrows (a subcat of the usual category of arrows) over the base.
I don’t understand, can you explain?
sheaves for a coverage, with its non-actual-pullback-stability condition, do form a Grothendieck topos.
oh, I see. That ruins that idea.
The category $Arr(C)$ is cofibred over $C$ when $C$ has pullbacks, but if we replace $Arr(C)$ with $Cov_J(C)$, which is the full subcat with objects the covers from the singleton coverage $J$, then the restriction $Cov_J(C) \to C$ is not quite cofibred, but is weakly so: we can always find lifts, but we don’t have the universal property, as in the case when pullbacks exist.
the axiom satisfied by coverages, […] That’s the essential axiom guaranteeing that sheaves form a topos.
Just to amplify, maybe:
One way to say this is: this is precisely the axiom that ensures that the reflector of the reflective subcategory of sheaves preserves finite limts, so that the inclusion of sheaves into presheaves becomes a geometric morphism of toposes.
Now, the category of presheaves of course has not just finite limits but many more good properties. So I suppose it is perfectly sensible to develop a notion of “nc toposes” or the like, where we say that an nc-topos is a reflective subcategory of a presheaf category that preserves some other property, not finite limits. I don’t know which property this might be, but it seems to me such a generalization is what the discussion here may be driving at.
Right, Urs, the nc sheafification is a localization as well but not left exact in general.
David: while the pullback of a cover to the localization is not a cover of localized category, one can look at interchanges order of functros as well and then it is OK. Yes, you have both maps in 16.
David, are you talking about the codomain fibration? That is fibered, not cofibered (although it is also an opfibration, but it doesn’t seem like that’s what you’re talking about); I think that’s what was confusing me. Given that, I see what you mean about non-cartesian lifts.
Right, Urs, the nc sheafification is a localization as well but not left exact in general.
Okay, so it might b good to characterize the properties that it does have.
Is nc-sheafifiation still left-adjoint to the inclusion of sheaves into presheaves?
And does it preserve finite products at least, maybe?
@Mike #23
opfibration
whoops, that’s what I meant.
Okay, so it might b good to characterize the properties that it does have.
I raised this question in 2007 to Rosenberg (at Arbeitstagung) who understands the subject of noncommutative sheafification the best, and he was enthusiastic in the first conversation, but the next day he came back with a grim opinion that he thinks there is no good Giraud-like characterization of such categories.
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