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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2018
    • (edited Jul 7th 2018)

    It’s still not quite right, is it? (here) After

    Moreover, up to equivalence, every Grothendieck topos arises this way:

    isn’t there the clause of accessible embedding missing? I.e. instead of

    the equivalence classes of left exact reflective subcategories PSh(𝒞)\mathcal{E} \hookrightarrow PSh(\mathcal{C}) of the category of presheaves

    it should have

    the equivalence classes of left exact reflective and accessivley embedded subcategories PSh(𝒞)\mathcal{E} \hookrightarrow PSh(\mathcal{C}) of the category of presheaves

    Or else, by the prop that follows, it should say

    the equivalence classes of left exact reflective and locally presentable subcategories PSh(𝒞)\mathcal{E} \hookrightarrow PSh(\mathcal{C}) of the category of presheaves


    (This is just a question. I didn’t make an edit. Yet.)

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 7th 2018
    • (edited Jul 7th 2018)

    It looks right to me as is. There should be an essential equivalence between left exact idempotent monads on a topos and Lawvere-Tierney topologies j:ΩΩj: \Omega \to \Omega, and if one has a Lawvere-Tierney topology in a presheaf topos, one ought to be recover a Grothendieck topology whose sheaves correspond to the algebras of the monad or objects orthogonal to jj-dense monomorphisms. These statements do not bring in accessibility conditions; I’d think accessibility of the subtopos embedding should be a consequence however.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 7th 2018

    See also Johnstone’s C.2.1.11:

    For a small category CC, the assignment JSh(C,J)J \mapsto Sh(C, J) is a bijection from the set of Grothendieck coverages on CC to the class of reflective subcategories of [C op,Set][C^{op}, Set] with cartesian reflector.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2018

    Okay, thanks!!

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 7th 2018

    added the additional pointer to Johnstone that Todd kindly provided. Also highlighted a bit more the remark (here) about the need to require accessibility (only) in the generality of \infty-toposes, lest I forget about this once again next time.

    diff, v4, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 7th 2018

    Yes, this is an interesting quirk. Is it known that accessibility is definitely necessary for \infty-toposes? I.e. is there an example of a non-accessible left exact reflective subcategory of an \infty-topos?

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