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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 8th 2010
    • (edited Jan 8th 2010)
    I think that the order of the definitions of a sieve should be swapped. The subfunctor definition is easier to state, more common (at least in my experience), and allows us to define descent in a total of two lines, which is equivalent to the other formulations of descent by the 2-Yoneda Lemma. That is, a grothendieck fibration F over a site is a prestack (resp. stack) provided that for any covering sieve S on an object U, the functor induced by the inclusion S -> h_U, Hom(h_U,F) -> Hom(S,F) is full and faithful (resp. an equivalence of categories). I'm not sure how we would define sieves in an (infinity,1)-categorical context, but this definition seems like it should generalize quite cleanly all the way up by the (infinity,1)-yoneda lemma with some minor tweaking.

    Edit: As an added upshot, this also gives us the sheaf condition for free by choosing F to be a discrete fibration.
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2010

    I guess I would be fine with this. I remember back when we created this entry, there was some discussion about what the best way to expose this is. I think opinions generally will differ, but it is certainly true that the subfunctor picture is the best one.

    Notice that there is closely related material also at sheaf and maybe there would be reason to try to merge and harmonize this more with that at sieve.

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 8th 2010
    Meanwhile, does anyone know of a good definition of an (infinity,1) sieve? Is there a chance that it is still a "subfunctor" of the "yoneda embedding of an object"? Can we formulate descent for a fibration of simplicial sets, or at least quasicategories using this very nice definition?
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2010

    This is a good question.

    I think indeed this is one of the very few places where there is something like a gap in Lurie's work:

    he doesn't have a notion of (oo,1)-site such that one would have the expected equivalence between oo-sheaves on the oo-site and geometric embeddings into the oo-presheaves.

    He does discuss in great detail how the structure of a 1-site on an oo-category may correspond to different notions of oo-sheaves: notably for each site there is the oo-topos of oo-sheaves that satisfy descent with respect to all Cech covers in the site. And then there is the hypercompletion of this, of oo-presheaves that satisfy descent also with respect to all hypercovers.

    Ideally one would think that there should be a notion of oo-site that encodes the difference between these.