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nLab > nLab General Discussions: Descent for categories of sheaves

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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 23rd 2018
    • (edited Jan 23rd 2018)
    Fix some site S.
    Consider the presheaf Sh on S valued in the bicategory of categories
    that sends s∈S to the category of sheaves over the slice site S/s.
    One can show that Sh is actually a sheaf of categories (or rather a stack in bicategories, to be precise).

    In other words, a sheaf can be specified locally on the elements of some cover and then glued together,
    and the same is true for morphisms of sheaves.

    Is this fact somewhere in the literature?
    I am also interested in the versions for ∞-sheaves on ∞-sites (any model will do).
    • CommentRowNumber2.
    • CommentAuthorJonasFrey
    • CommentTimeJan 23rd 2018

    I think a special case of this (for sheaves over spaces) is given as Exercise 2.7.D of Vakil’s Rising Sea. See also Remark 2.7.5.

    • CommentRowNumber3.
    • CommentAuthorMarc Hoyois
    • CommentTimeJan 23rd 2018

    This is Theorem 6.1.3.9 (3) in HTT.

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 26th 2018
    Thanks! Somehow I missed item (3) when looking through section 6.1.3.

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