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After Zoran had emphasized it for years without me ever really looking into it, now I have finally read the beginning of Kontsevich-Rosenberg’s article on “Q-categories” in more details… and was struck:
their notion of “generalized sheaves” is essentially nothing but the kind of condition that Lawvere considered in cohesive toposes $(u_! \dashv u^* \dashv u_* \dashv u^!) : T \to S$. More precisely, Lawvere considered the objects $x$ for which the canonical morphism $u_* x \to u_! x$ is an isomorphism. What Kontsevich-Rosenberg call generalized sheaves are those objects for which the other canonical morphism is an isomorphism: $u^* x \to u^! x$.
There are mainly two kinds of applications in Kontsevich-Rosenberg:
the original one was to find the right notion of sheaves over formal duals of non-commutative algebras. Apparently Rosenberg is fond of the insight that for a suitable cohesive presheaf topos (my words of course) the right condition is that $u^* x \to u^! x$ is an iso.
Apparently (if I remember correctly what Zoran told me) Kontsevich added the observation that formal smoothness and hence infinitesimal thickening is naturally described in this context. Now that I looked through it, I realize that what they talk about in this context is really pretty much exactly what I axiomatized as infinitesimal cohesion.
So I am happy: at once now the entire 79 page article by Kontsevich-Rosenberg turns out to be a great resource of examples and applications of cohesive topos technology! Notably they shed more light on the role of those infamous extra axioms that involve the two canoical natural transformations that come with any cohesive topos.
For that reason I have now begun expanding the $n$Lab entry Q-category that Zoran once started
Just for the record: while the $Q$-category of presheaves of sets on a $Q$-category is of course the (iso unit) adjunction between topoi, the category of sheaves inside the $Q$-category of presheaves is in general not a topos.
New stub unramified morphism with redirect formally unramfied morphism. It required so I also made a stub for the geometric and number-theoretic notion of residue field in the standard sense, and removed the redirect at field which has the other notion of the residue field in constructive mathematics. Both places have links to the other notion.
Urs, I think that Brzeziński is not working in the context of topoi but rather with abelian examples. I am just pointing to what you know: that one does not need to take presheaves of sets, but some other possibilities are interesting which allow for notion of formally smooth objects.
P.S. it is getting very interesting :)
the category of sheaves inside the Q-category of presheaves is in general not a topos.
And because of that, saying “sheaves” for the objects $X$ for which $u^* X \to u^! X$ is an iso is a really bad idea! :-)
Personally I would think these things should be given an entirely different name. But if we do stick with Kontsevich-Rosenberg’s terminology, we should at least be sure to say “$\mathbb{A}$-sheaves” or the like.
It is quite classical to use the sheaves with values in other categories, for example sheaves in abelian categories, then sheaves of Frechet algebras, which do not form an abelian category. Sheaf is in my opinion more about gluing conditions, not about the codomain category. I do not understand the pressure from topos community that if something is not a topos that this makes its objects not being sheaves, as the notion of a sheaf existed before the notion of a topos.
Personally I would think these things should be given an entirely different name.
But it HAS a different name. It is a sheaf on a Q-category with values in $C$, for example. Just spell the whole name. Resolutions existed in homological algebra for abelian categories, before the generalizations were found in nonabelian context. And people decided to still call them resolution, but they have a full name. The new name is indeed needed if it were the case that two different generalities in the same context would have the same name.
I agree with your remark about the $\mathbb{A}$-sheaves. In practice, the prefixes disappear if used a lot in the understood context.
I don’t mean the issue that the codomain may not be a topos. I mean that the $\mathbb{A}$-sheaf condition (saying that $u^* X \to u^! X$ is an iso) is not at all (in general) an ordinary sheaf condition.
But what would it mean an ordinary sheaf condition in that context ? In a number of cases it does have a feeling of a sheaf condition. Besides the sheafification is related to a localization at local morphisms and with correct notion of locality one reproduced the correct notion of sheafification, when the latter is possible, I think this is a strong argument for that terminology in this context. But never mind.
I just think that any terminology makes one say: “I have here a category of sheaves but – notice! – it is not a topos” is asking for trouble. Not every localization produces sheaves, and the terminology should try to faithfully reflect that.
But I can’t come up with a good term myself, currently.
You see, in noncommutative geometry we use sheaf intuition for a very long time, and the Q-category picture for the Zariski Q-category can be replaced in main examples by e.g. the quasi-site picture which is like a coverage, but one does not have the stability of the covers under pullbacks, and translated to such examples (which still have nothing to do with topoi, as well as the Frechet and abelian sheaves do not form topoi) one really have a sheaf-like conditions! Requiring to have a topos is extremely restrictive with respect to most noncommutative cases where I worked with sheaves in last 14 years, and I did NOT work with Q-categories but with other formalisms for abelian sheaves on noncommutative spaces. I worked with analogues of a small site, not of a big site, though.
By the way, it is not about arbitrary localization, not at all; but variosu generalizations of covers can indeed be inverted by talking the $\mathbb{A}$-sheaves for the corresponding choice of a Q-category $\mathbb{A}$. Of course, one should understand some intermediate generalities which are really interesting.
I have posted some discussion of this to the blog here
Zoran,
I am geting bit lost in the document, or maybe it’s too late: I want to extract concisely exactly the definition of a sheaf in noncommutative geometry, according to them.
Which definition (number?) has the covers or quasi-covers or whatever on $Alg_k^{op}$?
Ah, is it A.1.9.2?
First to answer your earlier question 6.5 explains that the classical notion of formal smoothness for MORPHISMS is as well as special case of the notion of formal smoothness with respect to a corresponding Q-category. Before that he explains the notion in the more usual language. Look also at the duality in part b) of Konst-Ros NcSp 6.5.
As far as A1.9.2 that is the flat topology on NAff. One can make variants adding corresponding finiteness conditions (like fppf and fpqc in commutative case). I am more used, however, to nc Zariski where one requires to work with flat localizations.
I like your new blog post, of course.
6.5 explains that the classical notion of formal smoothness for MORPHISMS is as well as special case of the notion of formal smoothness with respect to a corresponding Q-category.
I think in 6.5 no classical notion of formal smoothness appears. The statement is that formal smoothness with respect to any Q-category is equivalent to that with respect to some associated Q-category. No?
No.
Just one of the two is a $Q$-category. The notation has some meaning there. The other thing compared is the notion which is in a form like the classical notion.
The statement is a comparison of formal smoothness in terms of machinery in 6.3 (where one does not see any $Q$-category) and the formal smoothness in the sense of associated $Q$-category. It is straightforward to formulate the classical smoothness for morphisms in terms of 6.3 for the appropriate class of morphisms (remember that it is the same weather we require the lifting property with respect to affine thickenings, or any thickenings involving possibly nonaffine schemes). Then the 6.5. gives you the recipe for associating the corresponding $Q$-category, having the same notion of formal smoothness (etc.), exactly what you asked for.
Okay, as soon as I find a minute I look at this. Somebody should write this out cleanly in the nLab entry.
I expanded a bit smooth morphism of schemes. At least at one place there was a link to smooth morphism (which does not exist) which I changed to the link referring to smooth morphism of schemes. I do not know if you want a redirect smooth morphism at smooth morphism of schemes, or one wants to have a disambiguation entry at a future smooth morphism…
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