Zoran Skoda
discussion from point of a topos

Todd: How much does it matter? It matters if for example you want to say that points of Sh(X)Sh(X), XX a sober space, are in bijection with points of XX. Otherwise one can just refer back to equivalence of categories, unless you see a problem with that.

Mike Shulman?: I would argue that “a point” of a topos really should mean “a geometric morphism from SetSet,” not “an isomorphism class of geometric morphisms from SetSet,” for the same reason that “a group” means, well, “a group” and not “an isomorphism class of groups.” Following from that, I would say that it’s not really correct to say that points of Sh(X)Sh(X) (for XX sober) are in bijection with XX, but rather that the category of points of Sh(X)Sh(X) is equivalent to the category of points of XX. Note that that’s actually a stronger statement than saying that their sets of isomorphism classes of objects are in bijection.

Zoran Škoda: I want to use reconstruction theorems to get some geometric spaces; I need really to get points of underlying spaces without multiplicities! The equivalence is not satisfactory for my purposes, as I would like to use the (more general situations) in which one has some category TT of nice categories (e.g. abelian, topoi etc.) with a subcategory AA', where the morphisms are adjoint functors with possibly additional properties; possibly I want to pass to a comma category of the whole thing, for a specific object (the reasons for that are very specific and somewhat nontrivial, having to do with affinity of morphisms). Then I have an equivalence of categories between AA' and some category of local or test objects NAffNAff, which is in my examples some category of noncommutative algebras. Then I look at categories in TT which are obtained from gluing objects in AA', where gluing is via descent using say localizations with some flatness properties; this way I get some bigger category AA''. I do not assume that the localizations commute, i.e. the covers are more general than in the picture of Grothendieck topologies. Then I want to say that AA'' are represented by some class of presheaves on NAffNAff. For that I need to look at morphisms from objects in AA' to objects in AA'' without spurious multiplicity. Of course I can look at 2-Yoneda and getting some presheaf of categories on AA' and then afterwards try to decategorify to get down to a presheaf of sets on AA. I do not know what is the best approach. Any advice ?

Mike Shulman?: It sounds to me like you want to prove that the resulting (pseudo) presheaves of categories are essentially discrete, and hence are equivalent to presheaves of sets.

Urs Schreiber?: yes, I think, too, that this is what Zoran is talking about. I think effectively he has the setup discussed at notions of space only that there (,1)(\infty,1)-toposes are usesd in place where Zoran wants to use abelian categories, A A_\infty-categories and eventually stable \infty-categories as formal duals of spaces.

In that context, Mike: how do I see that the category [Set,T] geom[Set,T]_{geom} of geometric topos morphisms with natural transformations between them is equivalent to a set?

Mike Shulman?: It depends on what TT is. For an arbitrary topos TT, of course [Set,T] geom[Set,T]_{geom} will not be equivalent to a set. What sort of TT are you considering?

Zoran Škoda: My main examples are not in topos theory, but I would like to see the way similar proofs work. Instead of 2cat of topoi I need to consider certain slice 2cat of abelian categories. More precisely, start with a 2cat pCTpCT whose objects are pairs (a,O)(a,O) where aa is an abelian category and OO an object in aa; the morphisms are pairs of additive adjoint functors (no additional assumptions at start) together with maps Of *OO'\to f_* O. The slice category is over a category kModk-Mod where kk is a fixed unital ring, commutative or not, it does not matter. This is a ground category. The subcategory ApCTA'\subset pCT is given by the requirement that the pair of adjoint functors to the ground category is supposed to be affine (the right adjoint is faithful and has its own right adjoint). This forces the objects in sub2category AA' to be equivalent to RModR-Mod for some kk-ring RR; the fact that we are in 2-category means that the triangles in slice category commute up to isomorphisms, this nontrivially forces that the maps between two different RModR-Mod will not be general tensoring with a bimodule but really something coming from a ring map (affine morphisms satisfy such factorization conditions: similarly if cc is monadic over aa and bb over aa then cc is monadic over bb what is a special case of one of the adjoint lifting theorems; monadicity is weaker than affiness. In particular that means that in decategorified version (classes of geometric functors) the morphisms between categories of modules and underlying rings are the same (the Morita morphisms are excluded by the slice category trick). Now I glue such representable functors on NAff k=(kRings) opNAff_k = (k-Rings)^op like in gluing categories from localizations. I can assume that the cover is not only comonadic but in fact forms a noncommutative scheme of Rosenberg (plus that we work with choice of object OO not stated there, though automatic as inverse image of RR in kModk-Mod via the grounding morphism). Now I want to use some decategorification theorem to state that instead of gluing categories RModR-Mod I can glue representable presheaves h Xh_X with X=R opX = R^op; notice that localizations do not commute and the consecutive localizations do not form pullbacks, so we do not have stability axiom of Grothendieck topologies. I would like to be able to present all information on the glued category (noncommutative scheme) by a presheaf of sets on NAffANAff \cong A'; or understand if I really need presheaf of cats on NAffNAff. The strange locality given by localizations should give a subcategory of “sheaves” which is not a topos, but some subcategory of presheaves whose embedding into presheaves has weaker exactness conditions. Notice that while I glue representable presheaves on NAff, the consecutive (double) localizations where I compare them for gluing are NOT representable by objects in NAff, but only in the big ambient slice category of all abelian categories. In commutative case this may happen for nonsemiseparated schemes, but then we have still represent by the locally ringed spaces where we do not deal with 2-categories.

Zoran (P.S.) Mike, the main question for you before was if [Set,T] geom[Set,T]_{geom} is equivalent to a set when TT is a topos of sheaves over a topological space (the assertion is below in fact in a form of bijection which spurred the question). What or where is the proof ? (elephant?) P.S. 2 But I was asking all the time actually a different question, the domain is not Set but any of the members of a subcategory/family of local models. But I do not know good examples of such families in topoi (which have also decategorifications). P.S.3 Here is however an attempt for an example in Topoi but I am not sure if it is. Take the category Top of topological spaces. Then topological stacks are 1-stacks with some representability conditions; in particular they have an atlas by usual topological space. Now I do not know, but I suppose that the category of sheaves on a topological stack is still an elementary topos, though maybe not Grothendieck topos. Is it true that if I take [Sh(X),Sh(Y)] geom[Sh(X), Sh(Y)]_{geom} where XX is any topological space and YY a fixed topological stack, then this equivalent to a set? P.S. 4 Here is a further intuition. While the points of topoi are geom morphisms from Set, and Set is good enough to probe toplogical spaces, because they are made out of points, could not there be a more general statement that if one takes generalized S-points for S in some sub-2-category MODELS of Topoi which is equivalent to some 1-category, and if we look at topoi which are sheaf on some class of STACKS on MODELS possesing usual atlas conditions

(I want in the sense of gluing localization but to start with maybe gluing in Grothendieck topology is good starter)

are the S-points for all S in MODELS enough in 1-categorical sense ?

I said earlier topological stacks now “possessing usual atlas conditions” not just 1-stacks in usual sense because I need atlas to make sense of the category of sheaves on the stack.

Created on November 19, 2009 03:36:35 by Zoran Škoda (193.51.104.34)