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nLab > Latest Changes: Grothendieck fibration

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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeApr 2nd 2010

    Did some reorganizing at Grothendieck fibration.

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeApr 2nd 2010
    • (edited Apr 5th 2010)
    I was clicking through some of the links on that page, and I noticed something about prestacks, which are linked but for some reason not defined on the lab, for some reason. Are we (nLab) taking prestack to mean what Vistoli uses it to mean (i.e. a category fibered in groupoids that is separated with respect to the grothendieck topology [the canonical map given by the 2-yoneda lemma is injective faithful for any sieve S on U for all objects U in the base site]), or are we taking it to mean "a presheaf of groupoids", or even just a presheaf of categories?

    A question, as well: Why do algebraic geometers only consider stacks of groupoids?
    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeApr 2nd 2010

    I don't think "we" have used "prestack" enough to give it a definite meaning. I've always thought that "presheaf of groupoids/categories" would be a much better meaning of "prestack" than "separated presheaf of groupoids/categories" (where by "injective" I presume you mean "fully faithful") but the latter meaning seems to be universal especially in the algebraic geometry community, so I'd be hesitant to depart from it.

    Maybe an algebraic geometer can answer your second question better, but I think it's because a lot of the time those are what they see. For instance, if you take a sheaf of sets which is acted on by a sheaf of groups, or more generally an internal groupoid in sheaves, then its "homotopy quotient" is a stack of groupoids. I think most of the stacks that algebraic geometers care about are "moduli stacks" and arise in that way.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeApr 4th 2010
    • (edited Apr 4th 2010)

    Grothendieck construction approach to Baues-Wirsching cohomology can be found in the paper of Petar Pavesic, for whom I created an entry with a link to his (pale scan of the) paper in JPAA. The page is not created with the proper diacritics, which are however properly used inside the entry. It is open to improvements.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeApr 5th 2010

    Thanks, Zoran, that looks very helpful and goes some way to what I want. I knew Petar back when I used to visit Genoa quite often as he was often there.

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