Not signed in

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

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

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 internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology 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 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

1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeMar 31st 2020
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexI 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 <a href="https://ncatlab.org/nlab/revision/diff/simplicial+presheaf/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/simplicial+presheaf/17">v17</a>, <a href="https://ncatlab.org/nlab/show/simplicial+presheaf">current</a>

   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

   diff, v17, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeApr 9th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexDeleted 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-stack]]s 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](http://dx.doi.org/10.1090/s0002-9947-1973-0341469-9). * [[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](http://dx.doi.org/10.1007/bfb0067062). * [[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](http://dx.doi.org/10.1090/conm/055.1/862637) * [[J. F. Jardine]], _Simplical presheaves_. Journal of Pure and Applied Algebra 47:1 (1987), 35-87. [doi][1] [1]: http://dx.doi.org/10.1016/0022-4049(87)90100-9 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). <a href="https://ncatlab.org/nlab/revision/diff/simplicial+presheaf/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/simplicial+presheaf/19">v19</a>, <a href="https://ncatlab.org/nlab/show/simplicial+presheaf">current</a>

   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