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I edited three things: - In the definition of $D_X$ I corrected what seems to be a typo : indeed $D_X([n]) = X_n$ is a presheaf over $S$, 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 $C:=SSet^{S^{op}}$, but indeed $C^{op}\times C$
In the corollary that follows the introduction of $D_X$, I changed “evaluations of $X$” into “evalutations of $A$”, because in $[X,A]$, we end up with $holim lim A(U_i)$
Anonymous
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).
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