Timeline for Example of a locally presentable locally cartesian closed category which is not a topos?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 1, 2018 at 11:32 | comment | added | Mike Shulman | @MarcHoyois Yikes! Looks like that mistake is even my fault, and has been there for 8 years... gah. | |
| Mar 1, 2018 at 2:02 | comment | added | Marc Hoyois | @MikeShulman Ah OK, I was using the definition of Grothendieck quasitopos from this page, which seems wrong then. | |
| Feb 28, 2018 at 20:45 | comment | added | Mike Shulman | Also, a locale is actually a particular case of a Grothendieck quasitopos: it is the category of sheaves on itself that are separated for the maximal topology. | |
| Feb 28, 2018 at 20:28 | comment | added | Mike Shulman | Just a minor correction: not every Grothendieck quasitopos is "the category of separated presheaves on a site"; in general they consist of presheaves on some small category C that are separated for one topology on C and a sheaf for another topology on C. In other words, the separated objects for a Lawvere-Tierney topology on a Grothendieck (not necessarily presheaf) topos. | |
| Feb 28, 2018 at 20:24 | comment | added | Mike Shulman | FWIW, it is possible for a non-left-exact localization of a topos to be again a topos in its own right, even though the non-exact localization does not exhibit it as a (geometric) subtopos of the original one. Analogously, it should be possible for a localization without stable units of a topos to nevertheless happen to be a quasitopos in its own right. | |
| Feb 28, 2018 at 16:21 | comment | added | Marc Hoyois | My bad, actually the $\mathbb A^1$-localization does have stable units. Still, that doesn't imply it's an ∞-quasitopos, I don't think. | |
| Feb 28, 2018 at 15:59 | comment | added | Marc Hoyois | @TimCampion I suspect not but I don't know how one could rule it out. The reason I'm suspicious is that ∞-quasitopoi usually arise from localizations "with stable units" in the sense of Gepner-Koch, and the $\mathbb A^1$-localization does not have this property. On the other hand, motivic spaces have additional exactness properties that quasitopoi don't necessarily have, like disjoint coproducts, so really the only thing that fails is the effectivity of groupoid objects. | |
| Feb 28, 2018 at 6:04 | comment | added | Tim Campion | @MarcHoyois this begs the question -- is the unstable motivic category an $\infty$-quasitopos? i.e. is it the separated presheaves on a site? | |
| Feb 28, 2018 at 6:02 | comment | added | Tim Campion | Chalk this up as another one in the category of "I should have realized this"! I suppose it's not a total loss -- I've just added this class of examples to the nlab article on locally cartesian closed categories. | |
| Feb 28, 2018 at 5:55 | vote | accept | Tim Campion | ||
| Feb 28, 2018 at 5:53 | comment | added | David Roberts♦ | The category of diffeological spaces would be an example, then. | |
| Feb 28, 2018 at 5:40 | history | answered | Marc Hoyois | CC BY-SA 3.0 |