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To give an example of a peculiar feature of simplicial sets that I cannot remember encountering anywhere in the context of homotopy theory: every simplicial set $X$ possesses partial map classifier $X\rightarrowtail\widetilde X$: the $n$-simplices of $\widetilde X$ are partial simplices of $X$, i. e. maps from all kinds of simplicial subsets of the standard $n$-simplex $\Delta[n]$ to $X$. This $\widetilde X$ is a contractible Kan complex, in strongest possible way: it is an injective object, i. e. any $Y\leftarrowtail Y'\to\widetilde X$ extends to $Y$. It is thus some sort of a fibrant cone for $X$. This construction is functorial, in fact, part of a monad structure (sometimes called "lift monad" or "maybe monad"). Is not existence of such a thing useful for homotopy-theoretic purposes?

Note that this is just one example, there surely are many other features that can be extracted from the topos structure. I found a related question Internal logic of the topos of simplicial sets but it is rather about peculiarities of simplicial sets as a particular topos than peculiarities of this topos as a particular homotopy-theoretic universe.

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    $\begingroup$ Well, the theory of Cisinski model structure exploit the topos theoretic aspect to build the homotopy theory of simplicial sets. If not the partial arrow classifier some similar construction are used in the process (though I think we can do everything using only product with subobject classier). But it's not so clear that this specific construction has homotopy theoretic consequence (Beyond the infinity category of spaces being cartesian closed). $\endgroup$
    – Simon Henry
    Commented Feb 5, 2022 at 17:07

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To answer the title question: Yes, the fact that the 1-category $sSet$ is a 1-topos has homotopy-theoretic significance. It is closely related to the fact that the $\infty$-category $Spaces$ is an $\infty$-topos! To a large degree, constructions using the topos structure on $sSet$ can be recast model-independently as constructions using the $\infty$-topos structure on $Spaces$. If you like, I suppose you can see the fact that $Spaces$ is an $\infty$-topos as a consequence of the fact that $sSet$ is a 1-topos using Rezk's model topos machinery.

On the other hand, I'm not so sure about the significance of the partial map classifier in particular. The "$\infty$-partial map classifier" exists in $Spaces$, but there are not so many partial maps in the $\infty$-topos $Spaces$ as in the 1-topos $sSet$ (since a monomorphism is just a coproduct inclusion in $Spaces$). Probably this construction has some homotopy-theoretic significance in some model-dependent context, but it will depend on what the other model-dependent particulars of the situation are, I suspect.

On the gripping hand, an $\infty$-topos like $Spaces$ actually has object classifiers which are arguably even "better" than subobject classifiers.

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    $\begingroup$ mathoverflow.net/questions/372391/… $\endgroup$
    – Ivan Di Liberti
    Commented Feb 5, 2022 at 20:09
  • $\begingroup$ 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? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Feb 5, 2022 at 20:31
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    $\begingroup$ @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. $\endgroup$
    – Tim Campion
    Commented Feb 5, 2022 at 21:09
  • $\begingroup$ 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? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 6, 2022 at 2:42
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    $\begingroup$ @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. $\endgroup$
    – Tim Campion
    Commented Feb 6, 2022 at 3:50

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