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commented in the discussion at point of a topos and have a question there.
The discussion there is getting very off-topic; maybe it should be moved somewhere else. I don't really understand your last question, Zoran, but it seems to involve lots of specifics that don't have much to do with the general notion.
Specific or general, you used a lot of words that I don't understand and don't have time to understand. It wasn't at all clear to me that what I was being asked was about morphisms between topological spaces. A proof of that can be found at geometric morphism, although not dealing with the question of geometric transformations; the whole thing is in the Elephant, C1.4.5. In some context other than topological spaces, someone who understands that particular context (i.e. not me) would be better equipped to do a proof.
Okay, I think I understand the question better now.
The only question about whether sheaves on a topological stack is a topos is about size, right, because Top is large? If Top were a small category then so would any (small) stack be, and hence sheaves on such a stack would again be a Grothendieck topos. It seems fairly likely to me that the representability conditions on a topological stack would ensure that sheaves on it are equivalent to sheaves on some small site and hence a Grothendieck topos. I'd be pretty surprised if one could show that it was an elementary topos without showing that it's a Grothendieck topos.
I'm also pretty sure that geometric morphisms into sheaves on a topological stack will not, in general, be equivalent to a set, because the whole point of stacks is that they have automorphisms. For instance, I can consider some topological stacks presented by topological groupoids (i.e. internal groupoids in Top), and then every topological natural transformation between topological functors should induce a geometric transformation between corresponding topoi.
Can every topological stack be presented by a topological groupoid? If so, then there's a standard way to get a Grothendieck topos from a topological groupoid using equivariant sheaves, which seems like in good cases it would agree with sheaves on the stack in the other sense.
<blockquote>
A proof of that can be found at <a href="http://ncatlab.org/nlab/show/geometric+morphism">geometric morphism</a>, although not dealing with the question of geometric transformations; the whole thing is in the Elephant, C1.4.5.
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<p>Zoran, do you have the Elephant in reach at the moment? I don't I should look in the library today. About time that I start reading that book.</p>
<p>I suppose -- given the the proof at <a href="http://ncatlab.org/nlab/show/geometric+morphism">geometric morphism</a> which says that the geometric morphisms between localic toposes are as a set already in bijection with the set of continuous maps -- that there are no nontrivial natural transformations between these geometric morphisms in the first place.</p>
<p>This is the type of statement (if not its exact version) that I believe you want in your setup.</p>
<p>By the way, I'd be very interested in having an nLab page that discusses the similarity, as far as it goes, between</p>
<p>toposes <-> abelian categories</p>
<p>and then</p>
<p>(oo,1)-toposes <-> stable (oo,1)-categories .</p>
<p>I just don't feel that I understand this well enough myself to start such a page myself.</p>
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<p>Well, I still find <a href="http://www.noncommutative.org/#comment-92">David Ben-Zvi's message</a> seems to point in a useful direction.</p>
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Urs, The idea that categories of quasicoherent sheaves are analogs of topoi is certainly implicit in noncommutative algebraic geometry, but I don’t know where it’s explicit. However I wouldn’t say that one is a special case of the other, rather that one is a linearization of the other. Namely, rather than assigning to an algebra (or a scheme) the (oo-)category of sheaves of sets (or spaces) over it, such as those represented by other algebras, we are linearizing/stabilizing/Goodwillie-differentiating this assignment and considering sheaves of modules instead. Thus while of course you’re right that stable (oo,1)-categories are special cases of oo-topoi, the usual functor from schemes to the former is not the same as the usual functor to the latter, but rather is its stabilization.
A related point is that in my (highly unoriginal) view, NAG is not about algebras, but about these (oo-)categories of modules (ie algebras up to Morita equivalence) , so for example it seems misleading to me to look for topologies on the category of algebras, rather than on some category or 2-category of categories (say the oo-category of stable presentable oo-categories) in order to formulate NAG (though far smarter people disagree). (In particular I don’t see why non-accessible categories are relevant?) It’s also nice to note that most common categories are derived equivalent to modules over a (derived) algebra, so the difference mentioned in that n-lab page between NC algebraic geometry and NC topology is less pronounced in the derived setting.
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<p>Concerning the second bit: that's currently what you are talking about, Zoran, right? Replacing the site of algebras^op by a 2-cat of A-oo-categories?</p>
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I suppose -- given the the proof at geometric morphism which says that the geometric morphisms between localic toposes are as a set already in bijection with the set of continuous maps -- that there are no nontrivial natural transformations between these geometric morphisms in the first place.
If the space is separated enough, yes. But actually, and are (1,2)-categories via the specialization order, and it's that (1,2)-category that's equivalent to a full sub-2-category of .
Added to point of a topos two small previously missing assumptions: For geometric morphisms $Sh(Y) \to Sh(X)$ to bijectively correspond to continuous maps $Y \to X$, one needs that $X$ is sober; and for a Grothendieck topos $E$ to admit a geometric surjection from a topos of the form $Set/X$, one needs that $E$ has enough points.
Thanks! For the first, wouldn’t it be better to talk about locales?
Well, yes; but I think that pedagogically, it’s still nice to have the details spelled out in the context of topological spaces (as in the current form of the entry). I added the following comment:
“Note that the observation that the points of $Sh(X)$ are in bijection with the points of $X$ actually factors over an intermediate concept, namely that of points of a locale. Firstly, any topological space gives rise to a locale; if the space is sober, its points are in bijection with the locale-theoretic points of the induced locale. Secondly, for any locale (spatial or not), its locale-theoretic points correspond to the points of its induced sheaf topos.”
Thanks, that sounds great.
In the section “In sheaf toposes” what does “Con” in “ConFlatFunc” mean?
Presumably continuous functor.
A continuous functor, $C \to Set$, in MacLane-Moerdijk (p. 384)means that it sends covering sieves to colimit diagrams.
Maybe it would be clearer to write $CtsFlat$ instead of $ConFlat$.
Probably also need a reference to the warning at continuous functor.
I edited it to:
In sheaf toposes
The following characterization of points in sheaf toposes a special case of the general statements at morphism of sites.
Proposition 2.2
For $C$ a site, there is an equivalence of categories
$Topos(Set, Sh(C)) \simeq Site(C,Set) \,.$(Morphisms of sites $C\to Set$ are precisely the continuous flat functors.)
This appears for instance as (MacLaneMoerdijk, corollary VII.5.4).
Where “continuous” links to cover-preserving functor.
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