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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2010

    I had begun writing classifying topos of a localic groupoid and sheaves on a simplicial topological space, but am (naturally) being distracted from nLab work now, so this is left in somewhat unfinished form…

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 13th 2010

    The nomenclature “classifying topos of a localic groupoid”, although present in the title of that cited paper of Ieke’s, sounds just a tad odd to my ears. Or is it usual to refer to Set GSet^G as the classifying topos of a group GG?

    Yes, we can think of Set GSet^G as topos-theoretic cousin to the classifying space BGB G, but there is another cousin on the topos-theoretic side, namely Sh(BG)Sh(B G). They are a bit different, because for nice spaces XX the geometric morphisms Sh(X)Sh(BG)Sh(X) \to Sh(B G) correspond to continuous maps XBGX \to B G, whereas geometric morphisms Sh(X)Set GSh(X) \to Set^G correspond to GG-torsors in Sh(X)Sh(X), or equivalently to homotopy classes of continuous maps XBGX \to B G.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJul 13th 2010

    I don’t know whether it’s common to call Set GSet^G the “classifying topos of G.” It seems to me that tradition aside, it has as much right to be called that as the space BG does to be called the “classifying space of G” – in both cases what is actually being classified are G-torsors. But perhaps for that reason “the classifying space of G” was a confusing term from the beginning and we shouldn’t compound the problem.

    Do I remember correctly that Set GSet^G and Sh(BG)Sh(BG) are actually “weak homotopy equivalent” as toposes?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010

    Do I remember correctly that Set GSet^G and Sh(BG)Sh(B G) are actually “weak homotopy equivalent” as toposes?

    Yes. This is discussed and proven in Ieke’s Springer Lecture notes.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010

    It’s theorem 1.1, proven from page 77 on.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2017

    I have

    1. expanded a tad at classifying topos of a localic groupoid,

    2. made classifying topos of a topological groupoid a redirect to it,

    3. added to Grothendieck topos a brief remark under Properties – As localic groupoids (since any such mentioning had been missing there).

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