Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeNov 28th 2011

    Every category – indeed, every simplicial set – admits a homotopy final functor into it out of a Reedy category, namely its category of simplices (HTT This makes me wonder: can every (,1)(\infty,1)-topos be presented as a localization of an (,1)(\infty,1)-topos of presheaves on a Reedy category?

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 28th 2011

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

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 29th 2011

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

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeNov 29th 2011

    And I guess with that point made, it makes sense to ask the question more generally about locally presentable (,1)(\infty,1)-categories. I’m thinking of something like this: suppose C is a small (,1)(\infty,1)-category and (ΔC)(\Delta\downarrow C) its category of simplices; then we have a functor t:(ΔC)Ct\colon (\Delta\downarrow C) \to C sending each simplex to the last object occurring in it. This induces a functor t *:sPre(C)sPre(ΔC)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 (ΔC)(\Delta\downarrow C) which fix the last object. Consider the localization of sPre(ΔC)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)sPre(C)?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeDec 10th 2011

    At least in the 1-categorical case, this is true. The functor t *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)Pre(C) as a reflective subcategory of Pre(ΔC)Pre(\Delta\downarrow C). Does C3.3.8(i) have an (,1)(\infty,1)-categorical analogue?