Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
  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