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    • CommentRowNumber1.
    • CommentAuthorUlrik
    • CommentTimeOct 30th 2018

    Page created, but author did not leave any comments.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUlrik
    • CommentTimeOct 30th 2018

    I added a stubby version of (∞,1)-comparison lemma with a version by Hoyois. We should also have pages for the (∞,1)-versions of comorphism of sites and morphism of sites.

    I was wondering if there’s any useful generalization not requiring a lot of pullbacks to exist in the (∞,1)-sites? Of course, we have to be careful as this discussion shows.

    Is there some property of an (∞,1)-site that guarantees that the sheaf topos has finite homotopy dimension, and could we combine such a property with the usual comparison lemma to get a different, general (∞,1)-comparison lemma?

    • CommentRowNumber3.
    • CommentAuthorMarc Hoyois
    • CommentTimeOct 30th 2018

    I made a few adjustments (in particular a locally small assumption on DD is missing from the published version of my paper).

    I never found a useful generalization that does not assume existence of some pullbacks. For example there are obvious formulations of a and c that do not require pullbacks (let’s assume DD small for simplicity): a. for every τ\tau-sieve UXU\to X, a ρu !(UX)a_\rho u_!(U\to X) is an equivalence; c. for every XDX\in D, its image in Shv ρ(D)Shv_\rho(D) belongs to the smallest subcategory generated by the image of CC under colimits. Then a+b+c are necessary and sufficient for the conclusion of the lemma. Condition a is also what the definition of “continuous functor” should be in the (∞,1)-context (b is of course the definition of “cocontinuous functor”).

    • CommentRowNumber4.
    • CommentAuthorUlrik
    • CommentTimeOct 30th 2018
    • (edited Oct 30th 2018)

    Thanks for your comments and adjustments!

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 4th 2019

    Cross-linked to dense subsite.

    diff, v4, current