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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeNov 28th 2011
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   Author: Mike Shulman
   Format: MarkdownItexEvery 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?

   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 -topos be presented as a localization of an -topos of presheaves on a Reedy category?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 28th 2011
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   Author: Urs
   Format: MarkdownItexYou mean by a possibly non-lex localization, right? (Because by [this proposition](http://ncatlab.org/nlab/show/n-localic+%28infinity,1%29-topos#Examples) lex localizations of $\infty$-presheaves over 1-sites are 1-localic. )

   You mean by a possibly non-lex localization, right? (Because by this proposition lex localizations of -presheaves over 1-sites are 1-localic. )

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeNov 29th 2011
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   Author: Mike Shulman
   Format: MarkdownItexI guess I would have to mean that, wouldn't I? (-:

   I guess I would have to mean that, wouldn’t I? (-:

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeNov 29th 2011
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   Author: Mike Shulman
   Format: MarkdownItexAnd 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 $sPre(C)$?

   And I guess with that point made, it makes sense to ask the question more generally about locally presentable -categories. I’m thinking of something like this: suppose C is a small -category and its category of simplices; then we have a functor sending each simplex to the last object occurring in it. This induces a functor , and every object in the image of this functor has the property that it sees as isomorphisms all the maps in which fix the last object. Consider the localization of which forces all these maps to be invertible; it seems as though that has a decent chance to be equivalent to ?

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeDec 10th 2011
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   Author: Mike Shulman
   Format: MarkdownItexAt 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?

   At least in the 1-categorical case, this is true. The functor has a left adjoint (left Kan extension), and by C3.3.8(i) in the Elephant, it is fully faithful; thus it exhibits as a reflective subcategory of . Does C3.3.8(i) have an -categorical analogue?