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Every category – indeed, every simplicial set – admits a homotopy final functor into it out of a Reedy category, namely its category of simplices (HTT 4.2.3.14). This makes me wonder: can every $(\infty,1)$-topos be presented as a localization of an $(\infty,1)$-topos of presheaves on a Reedy category?
You mean by a possibly non-lex localization, right? (Because by this proposition lex localizations of $\infty$-presheaves over 1-sites are 1-localic. )
I guess I would have to mean that, wouldn’t I? (-:
And I guess with that point made, it makes sense to ask the question more generally about locally presentable $(\infty,1)$-categories. I’m thinking of something like this: suppose C is a small $(\infty,1)$-category and $(\Delta\downarrow C)$ its category of simplices; then we have a functor $t\colon (\Delta\downarrow C) \to C$ sending each simplex to the last object occurring in it. This induces a functor $t^* \colon sPre(C) \to sPre(\Delta\downarrow C)$, and every object in the image of this functor has the property that it sees as isomorphisms all the maps in $(\Delta\downarrow C)$ which fix the last object. Consider the localization of $sPre(\Delta\downarrow C)$ which forces all these maps to be invertible; it seems as though that has a decent chance to be equivalent to $sPre(C)$?
At least in the 1-categorical case, this is true. The functor $t^*$ has a left adjoint (left Kan extension), and by C3.3.8(i) in the Elephant, it is fully faithful; thus it exhibits $Pre(C)$ as a reflective subcategory of $Pre(\Delta\downarrow C)$. Does C3.3.8(i) have an $(\infty,1)$-categorical analogue?
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