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Tim Hosgood (Sep 23 2021 at 14:35):we can consider “the category of (sheaves on manifolds)”, but also “the category of (sheaves on (the category of manifolds))”. it seems to me (with my poor intuition) that the latter should sit inside the former somehow: manifolds are “just” bits of euclidean space glued together, so if we glue them together then that’s the same as just gluing bits of euclidean space together, and this gives (locally free) sheaves on manifolds
Tim Hosgood (Sep 23 2021 at 14:36):i suppose there are two more general questions i’m asking here: is there a relation between sheaves on a category and sheaves on the objects of that category; and is “taking the category of sheaves” something sort of idempotent?
Tim Hosgood (Sep 23 2021 at 14:37):(a key point i suppose is that arbitrary sheaves on manifolds can just be sheaves of sets, and i’m really only interested in the “geometric” ones, ie ones that locally look like euclidean space)
Tim Hosgood (Sep 23 2021 at 14:37):(so i suppose something more general than just locally free sheaves, since i’m not asking for constant rank)
Morgan Rogers (he/him) (Sep 23 2021 at 14:42):Tim Hosgood said:
i suppose there are two more general questions i’m asking here: is there a relation between sheaves on a category and sheaves on the objects of that category; and is “taking the category of sheaves” something sort of idempotent?
This sounds like the idea of gros vs. petit toposes, but what you wrote in the preceding message seems a little strange to me, since I've not really considered gluing all of the sheaf toposes together (which I presume is what you're suggesting with "the category of (sheaves on manifolds)")
Tim Hosgood (Sep 23 2021 at 14:44):oh that’s true, i suppose it is odd to not fix one specific base manifold maybe…
Reid Barton (Sep 23 2021 at 14:45):Right, I assume one of your categories is: take the category Man of manifolds and give it the coverage whose covers are jointly surjective families of open embeddings, and take the category of sheaves for that. But I'm not sure which one this is and what the other one is
Tim Hosgood (Sep 23 2021 at 14:45):but yes, i think i still need to internalise this idea of gros vs petit
Tim Hosgood (Sep 23 2021 at 14:45):that’s the latter one
Tim Hosgood (Sep 23 2021 at 14:45):Sh(Man)
Tim Hosgood (Sep 23 2021 at 14:48):i suppose another question is: am i right in thinking that we can recover the category Man as Sh(Eucl), or something like this?
Reid Barton (Sep 23 2021 at 14:49):So some of these sheaves "are" just manifolds again, in the sense that they're representable, even if we might have presented them by gluing together several manifolds. But there will be a lot of other sheaves that we can build that aren't manifolds, for instance because Sh(Man) is cocomplete but Man isn't. For example, we could form pushouts along closed embeddings or quotients by strange equivalence relations.
Reid Barton (Sep 23 2021 at 14:51):Tim Hosgood said:
i suppose another question is: am i right in thinking that we can recover the category Man as Sh(Eucl), or something like this?
Something like this, but there are restrictions on what kinds of gluings you can perform (again for instance because Sh(Eucl) is cocomplete, and Man is not).
Reid Barton (Sep 23 2021 at 14:52):I don't know if it will answer your questions, but you could take a look at https://ncatlab.org/nlab/show/Master+course+on+algebraic+stacks.
Matteo Capucci (he/him) (Sep 23 2021 at 14:53):According to the nLab, Sh(Eucl) is equivalent to Sh(Diff) which is the topos of smooth spaces
Tim Hosgood (Sep 23 2021 at 14:54):oh I’ve been meaning to read those notes by Toën for a long time now!
Reid Barton (Sep 23 2021 at 14:54):The first two sections of "Master course..." are about this example of viewing manifolds as a certain kind of sheaves on Eucl, and other analogous situations.
Tim Hosgood (Sep 23 2021 at 14:54):ah that makes sense, you get something strictly more general than just manifolds
Tim Hosgood (Sep 23 2021 at 14:55):what about my other question? Sh(Sh(C)) compared to Sh(C)
Tim Hosgood (Sep 23 2021 at 14:55):(since i’m getting such great answers i might as well ask for more 😉)
Reid Barton (Sep 23 2021 at 15:08):These kinds of constructions are all of the following general form: we adjoin certain kinds of colimits (e.g. all colimits, or just ones of the kind that result in manifolds) while preserving certain existing colimits (e.g. the ones that present a manifold in terms of an open cover). So if you compose two of these and the stars align, you will get another construction of the same kind. It might not be very easy to make this precise.
Reid Barton (Sep 23 2021 at 15:10):In the case of Sh(Sh(C)) = Sh(C), any topos E has a canonical topology whose category of sheaves is again equivalent to E. (A priori there could be size issues in forming Sh(E), but actually they aren't because this category turns out to be equivalent to E.)
Tim Hosgood (Sep 23 2021 at 15:15):oh, yes! i vaguely knew this at one point but had forgotten, thank you :)
Fawzi Hreiki (Sep 24 2021 at 00:33):I'm not sure what the answer is for manifolds, but I remember reading a while ago about how you can get the big and little Zariski toposes from each other
Fawzi Hreiki (Sep 24 2021 at 00:37):Shouldn't the full and faithful functor induce a subtopos inclusion from the small topos of to the big one.
Fawzi Hreiki (Sep 24 2021 at 00:40):Composing with the forgetful functor then should give a geometric morphism from the small topos of to the big topos of the point manifold. However, this won't necessarily be subtopos inclusion since different open subspaces can be isomorphic just as spaces.
Reid Barton (Sep 25 2021 at 11:32):Matteo Capucci (he/him) said:
According to the nLab, Sh(Eucl) is equivalent to Sh(Diff) which is the topos of smooth spaces
I happened to come across the beginning of section C.2.2 of the Elephant, which discusses situations in which a "dense" subsite of a site (like Eucl inside Diff here) has the same category of sheaves.
Fawzi Hreiki (Sep 25 2021 at 12:50):Yes but the subcategory of open subspaces of a chosen manifold won’t be dense in manifolds
Fawzi Hreiki (Sep 25 2021 at 12:50):Only the open subspaces of all Cartesian spaces
Morgan Rogers (he/him) (Sep 25 2021 at 13:44):Reid Barton said:
Matteo Capucci (he/him) said:
According to the nLab, Sh(Eucl) is equivalent to Sh(Diff) which is the topos of smooth spaces
I happened to come across the beginning of section C.2.2 of the Elephant, which discusses situations in which a "dense" subsite of a site (like Eucl inside Diff here) has the same category of sheaves.
For full generality, there's Caramello's paper on "Denseness conditions", which also provides conditions on morphisms and comorphisms of sites to produce other kinds of geometric morphism