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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)

    I am wondering about the following question, concerning 2-sheaves:

    Let \mathcal{B} be a sheaf-topos. By Bunge-Pare, cor 2.6 every geometric morphism

    f: f : \mathcal{E} \to \mathcal{B}

    into \mathcal{B} is a 2-sheaf on \mathcal{B} equipped with the canonical topology, in the sense that the corresponding indexed category

    𝒞 /f *(): opCat \mathcal{C}_{/f^*(-)} : \mathcal{B}^{op} \to Cat

    is a 2-sheaf.

    Moreover, by inspection and by a standard fact, if ff is an etale geometric morphism, then this 2-sheaf is actually a 1-sheaf, equivalently: is represented by an object of \mathcal{B}.

    So we have this sequence of 2-functors

    Sh 1()(Topos /) etTopos /Sh 2(). \mathcal{B} \simeq Sh_{1}(\mathcal{B}) \simeq (Topos_{/\mathcal{B}})_{et} \hookrightarrow Topos_{/\mathcal{B}} \hookrightarrow Sh_2(\mathcal{B}) \,.

    Unless I am missing something.

    Is the last inclusion that of 2-sheaves which factor through the forgetful functor ToposCatTopos \to Cat?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeMar 9th 2012

    Yes, that’s right. Except there is a size switch: \mathcal{B} is equivalent to the category of small 1-sheaves on itself, whereas the indexed category corresponding to a geometric morphism is a large 2-sheaf. In particular, the composite Sh 1()Sh 2()Sh_1(\mathcal{B}) \to Sh_2(\mathcal{B}) is not the obvious inclusion of 1-sheaves in 2-sheaves.

    I don’t understand your last question.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeMar 9th 2012
    • (edited Mar 9th 2012)


    In particular, the composite Sh 1()Sh 2()Sh_1(\mathcal{B}) \to Sh_2(\mathcal{B}) is not the obvious inclusion of 1-sheaves in 2-sheaves.

    I noticed this while running to the train. It sends an object XX \in \mathcal{B} to the functor A /X×AA \mapsto \mathcal{B}_{/ X \times A}. So how should we think of this inclusion?

    I don’t understand your last question.

    For a 2-sheaf opCat\mathcal{B}^{op} \to Cat we can ask if it is equivalent to a composite opToposCat\mathcal{B}^{op} \to Topos \to Cat.

    I was wondering if those that have this factorization are those in the image of the last inclusion.

    (I should be able to figure that out now. Currently I am throwing out these questions while travelling…)

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMar 12th 2012

    So how should we think of this inclusion?

    I would say, Sh 1(B)Sh_1(B) is a “category of sets” and Sh 2(B)Sh_2(B) is a “2-category of categories”, and this functor sends each “set” XX to the “overcategory” of “sets” over XX.