Timeline for Is there a concrete application of topos theory?
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27 events
| when toggle format | what | by | license | comment | |
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| Apr 3 at 2:12 | answer | added | rpk | timeline score: 3 | |
| S Apr 26, 2023 at 5:00 | history | bounty ended | CommunityBot | ||
| S Apr 26, 2023 at 5:00 | history | notice removed | CommunityBot | ||
| Apr 21, 2023 at 5:37 | comment | added | Ian Agol | Topos theory is being considered at this site: topos.institute | |
| S Apr 18, 2023 at 3:06 | history | bounty started | user234212323 | ||
| S Apr 18, 2023 at 3:06 | history | notice added | user234212323 | Draw attention | |
| Dec 1, 2022 at 15:29 | comment | added | Simon Henry | Any morphisms of topos can be described using well chosen sites - especially if you used functors a site to the category of sheaves, and you can always work with pretopologies if you want to - what you are saying apply to literally everything in topos theory. | |
| Dec 1, 2022 at 15:20 | comment | added | R. van Dobben de Bruyn | @SimonHenry I don't agree at all that there is no difference between topoi and sites. On objects that might be correct, but the point is that most morphisms of topoi you encounter in algebraic geometry come from a functor of sites with some additional properties. My experience as a working algebraic geometer (and please believe me when I say this) is that people learn étale cohomology using pretopologies (not even topologies), and for many it will not be clear what the notion of "morphism of topoi" buys you that you couldn't already do using geometry instead of category theory. | |
| Dec 1, 2022 at 2:33 | comment | added | Simon Henry | @R.vanDobbendeBruyn I honestly don't understand the point you want to make. There is literally no difference between studying Grothendieck toposes and studying sites except the name you give to things. A more accurate comparison is someone asking you what are the application of the notion of manifold - and then saying that "yes, there is such and such example of applications, but for each of them I can get around using explicit coordinate system" - I mean, sure, you can always do that - but that's basically the whole point of manifolds... | |
| Dec 1, 2022 at 0:58 | comment | added | R. van Dobben de Bruyn | @SimonHenry and AndresBlass: do you also compute limits (I mean of real numbers) using filters? That's how Bourbaki sets it up, and it's essentially the same as what we teach in calculus. But most working algebraic geometers need to talk about sieves or topoi just as much as calculus students need to talk about filters. It's a lot of definitions and machinery, but I think the question of what this buys you is not only a well-defined one, but also a sensible one. | |
| Jan 20, 2021 at 11:02 | comment | added | Kim | @PhilHarmsworth Skimming through it, I didn't see anything that looked like an essential application. | |
| Jan 20, 2021 at 8:00 | comment | added | Phil Harmsworth | Are the examples listed here oliviacaramello.com/Unification/ConcreteExamples.html the kind of thing you're looking for? | |
| Jan 16, 2021 at 21:32 | comment | added | Kim | @Faris I like the sound of that. Is this proven in Berthelot-Ogus (or some other standard reference)? One (possibly related?) property I would be interested to see used to do something concrete is the fact that a topos can have more morphisms than it would get from being represented by any single site. | |
| Jan 16, 2021 at 21:28 | comment | added | Kim | @DenisNardin Um, maybe not. I wouldn't consider a spectral version of anything to be "concrete", especially given that it is building topoi into the foundations. Using topoi to prove things about topoi is what I'm trying to avoid. | |
| Jan 16, 2021 at 16:58 | comment | added | Denis Nardin | The definition of spectral stacks use the notion of locally ringed (higher) topos as a replacement for the the notion of locally ringed space. Is this the kind of thing you are looking at (this is a definition and not a theorem, but of course any result about spectral DM-stacks uses their definition). | |
| Jan 16, 2021 at 16:12 | comment | added | Faris | @Kim I think you can't lift the functoriality of the crystalline topos to sites so that might be an example. | |
| Jan 16, 2021 at 15:55 | comment | added | Kim | @AndreasBlass One thing that I would like to see, if it exists, is a concrete result where topos theory is "used" because multiple sites with the same topos show up, and it gets so confusing to keep track of individual sites that the topos is the only reasonable thing to think about. | |
| Jan 16, 2021 at 15:50 | comment | added | Andreas Blass | Since many different sites can give the same topos of sheaves, I'm inclined to view a topos as capturing the "important" aspects of a site and discarding irrelevant details. I could describe this as an analogy "topos : site :: group : group-presentation. In principle, group theory could be developed entirely in terms of presentations, never mentioning the groups themselves, and in some situations that's useful, but in most situations it just makes things less clear. | |
| Jan 16, 2021 at 15:13 | comment | added | Simon Henry | Yes "Grothendieck topos theory" and "site theory" are two different point of view on exactly the same thing. Anything you can do with one you can also do it with the other. Of course there are indeed a lot of technical and conceptual advantages of working with toposes rather than sites, but at the end of the day you can always translate everything in terms of sites. | |
| Jan 16, 2021 at 15:12 | comment | added | Dmitri Pavlov | @Kim: I have never seen the term "site theory" in the literature. Topos theory could also be called sheaf theory, but this term has already been used for a special subarea, the theory of sheaves of abelian groups and chain complexes. Accordingly, a new name was necessary for sheaves of sets and other nonabelian objects. For instance, geometric morphsism of toposes involving functors like f^* and f_* are used all the time in fields like motivic homotopy theory. | |
| Jan 16, 2021 at 15:02 | comment | added | Kim | @SimonHenry I guess what I mean is the notion of topos is not used. Because it is highlighted in SGA 4 with its own special name and accompanying foundational results, I read this as implicitly suggesting that the notion itself should be an important conceptual landmark, above and beyond that of a (sheaf on a) site. But then Deligne's expositions of the Weil conjectures avoid the term entirely, which makes me think it is extraneous for that purpose. Am I mistaken? Should I always read (Grothendieck) "topos theory" as just a fancy way of saying "site theory" and nothing more? | |
| Jan 16, 2021 at 14:58 | comment | added | Simon Henry | I'm confused by how you consider that topos theory is not used in the proof of the Weil conjecture. You seem to be saying that there use can be replaced by Grothendieck topologies and étale cohomology (which I agree with) but topos theory as introduce in SGA 4 is really nothing more than the theory of Grothendieck topologies. So I wouldn't consider this as getting ride of topos theory. I'm pointing this out, because the same thing apply to all application of Grothendieck toposes : you can always write everything in terms of Sites and Grothendieck topologies. | |
| Jan 16, 2021 at 14:20 | comment | added | Kim | One could also, in principle, reduce any proof using sheaves into a proof not using sheaves, if one is willing to replace every occurrence of sheaf or sheaf cohomology with a horribly complex system of open covers and intersections, etc. But I think this is not the spirit of reduction I want to allow. It's better to say that sheaves are "essential" in this case. The same for topos theory. It can be honestly reduced out of etale cohomology. But maybe there is something else where "reducing it out" gives you something, not just dirtier, but impossible to follow conceptually. | |
| Jan 16, 2021 at 14:15 | comment | added | David Roberts♦ | I suspect the answer might be 'no', since one can always beta-reduce a proof to avoid referring to a topos (much like people talk about étale cohomology without mentioning the étale topos). But maybe there are examples where the topos viewpoint gives a cleaner interpretation of the proof, or a cleaner proof? | |
| Jan 16, 2021 at 14:12 | comment | added | Kim | Anything really. I imagine algebraic geometry and number theory are the most likely suspects, but if you know a neat fact about curvature of manifolds or solutions of a PDE that can only be obtained by using a topos somewhere, I would love to hear it. Proving internal results about category theory, or topos theory (or higher versions thereof) don't count. | |
| Jan 16, 2021 at 13:42 | comment | added | David Roberts♦ | A concrete result in which field? Algebraic geometry? Differential geometry? Any field that's not category theory? (Genuinely curious, since you haven't used a top-level tag) | |
| Jan 16, 2021 at 12:20 | history | asked | Kim | CC BY-SA 4.0 |