Timeline for Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?
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| Feb 6, 2022 at 3:50 | comment | added | Tim Campion | @R.vanDobbendeBruyn Sure, Giraud axioms are one characterization -- see HTT 6.4.1.5. "Locally cartesian closed + object classifiers" or "locally cartesian closed + descent" are others -- see HTT 6.1.6.8. Unlike "left exact localization of a presheaf topos", it's not trivial that $Spaces$ has these properties. | |
| Feb 6, 2022 at 2:42 | comment | added | R. van Dobben de Bruyn | I'm confused by what you write about $\mathscr S$ being an $\infty$-topos. In the definition Lurie gives (HTT, Def. 6.1.0.4) this is trivially true. Are you thinking of a different definition, e.g. "Giraud's axioms"? What is the key thing to check? | |
| Feb 5, 2022 at 21:09 | comment | added | Tim Campion | @მამუკაჯიბლაძე I'm not sure. The closest thing I'm sure I've seen is stuff about object classifiers, for which the main reference is in HTT (as indicated on the nlab page). Maybe something along the lines you're asking for would be the newer constructions of straightening / unstraightening (for left and right (i.e. "discrete") fibrations this is in Cisinski's book, and in general it's work of Hoang Kim Nguyen, but the idea goes back to work of Voevodsky related to homotopy type theory. Here, topos-y properties of $sSet$ are used to model the $\infty$-categorical Grothendieck construction. | |
| Feb 5, 2022 at 20:31 | comment | added | მამუკა ჯიბლაძე | Very interesting! So the analog of $\widetilde X$ must be something like $\Sigma_{S:U}X^S$ where $U$ is some universe. Does this appear somewhere? | |
| Feb 5, 2022 at 20:09 | comment | added | Ivan Di Liberti | mathoverflow.net/questions/372391/… | |
| Feb 5, 2022 at 19:36 | history | answered | Tim Campion | CC BY-SA 4.0 |