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nLab > Latest Changes: sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes

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  1.  
    • 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 ℰ↪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 ℰ↪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 ℰ↪PSh(𝒞)\mathcal{E} \hookrightarrow PSh(\mathcal{C}) of the category of presheaves

    No?

    (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:Ω→Ωj: \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 J↦Sh(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?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 16th 2021

    The remark (here) on topological localizations being accessible pointed to HTT Prop. 6.2.1.5. I have changed that to read “Cor. 6.2.1.6”, where the statement is actually made explicit.

    diff, v6, current

  8. I found the references impossible to follow, so I wrote my own.

    Shane

    diff, v7, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 5th 2022

    Thanks. Each time I want to reference this fact, I keep feeling we once had a detailed discussion of it on the nnLab somewhere, only to not find it. Maybe we never did.

    diff, v8, current

1 to 9 of 9

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