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  1. I edited three things: - In the definition of D XD_X I corrected what seems to be a typo : indeed D X([n])=X nD_X([n]) = X_n is a presheaf over SS, not a set (and then a simplicial presheaf, not a simplicial set)

    • In the definition of the notation [,][-,-] for simplicial presheaves, I edited the domain of that functor, which is not simply C:=SSet S opC:=SSet^{S^{op}}, but indeed C op×CC^{op}\times C

    • In the corollary that follows the introduction of D XD_X, I changed “evaluations of XX” into “evalutations of AA”, because in [X,A][X,A], we end up with holimlimA(U i)holim lim A(U_i)


    diff, v17, current

    • CommentRowNumber2.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 9th 2021

    Deleted this:

    The theory of simplicial presheaves and of simplicial sheaves was developed by J. F. Jardine in a long series of articles, some of which are listed below. It’s usage as a model for infinity-stacks was developed by Toën as described at infinity-stack homotopically.

    Added this:

    The original articles are

    • Kenneth S. Brown, Abstract homotopy theory and generalized sheaf cohomology. Transactions of the American Mathematical Society 186 (1973), 419-419. doi.

    • Kenneth S. Brown, Stephen M. Gersten, Algebraic K-theory as generalized sheaf cohomology. In: Higher K-Theories. Lecture Notes in Mathematics (1973), 266–292. doi.

    • J. F. Jardine, Simplicial objects in a Grothendieck topos. In: Applications of algebraic K-theory to algebraic geometry and number theory. Contemporary Mathematics (1986), 193-239. doi

    • J. F. Jardine, Simplical presheaves. Journal of Pure and Applied Algebra 47:1 (1987), 35-87. doi A modern expository account is

    • John F. Jardine, Local Homotopy Theory, Springer, 2015. [doi](Applications of algebraic K-theory to algebraic geometry and number theory).

    diff, v19, current

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