nLab
simplicial presheaf

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Definition

Simplicial presheaves over some site SS are

  • Presheaves with values in the category SimpSet of simplicial sets, i.e., functors S opSimpSetS^{op} \to \Simp\Set, i.e., functors S op[Δ op,Set]S^{op} \to [\Delta^{op}, \Set];

or equivalently, using the Hom-adjunction and symmetry of the closed monoidal structure on Cat

  • simplicial objects in the category of presheaves, i.e. functors Δ op[S op,Set]\Delta^{op} \to [S^{op},\Set].

Interpretation as \infty-stacks

Regarding SimpSet\Simp\Set as a model category using the standard model structure on simplicial sets and inducing from that a model structure on [S op,SimpSet][S^{op}, \Simp\Set] makes simplicial presheaves a model for \infty-stacks, as described at infinity-stack homotopically.

In more illustrative language this means that a simplicial presheaf on SS can be regarded as an \infty-groupoid (in particular a Kan complex) whose space of nn-morphisms is modeled on the objects of SS in the sense described at space and quantity.

Examples

  • Notice that most definitions of \infty-category the \infty-category is itself defined to be a simplicial set with extra structure (in a geometric definition of higher category) or gives rise to a simplicial set under taking its nerve (in an algebraic definition of higher category). So most notions of presheaves of higher categories will naturally induce presheaves of simplicial sets.

  • In particular, regarding a group GG as a one object category BG\mathbf{B}G and then taking the nerve N(BG)SimpSetN(\mathbf{B}G) \in \Simp\Set of these (the “classifying simplicial set of the group whose geometric realization is the classifying space G\mathcal{B}G), which is clearly a functorial operation, turns any presheaf with values in groups into a simplicial presheaf.

Remarks

Properties

Here are some basic but useful facts about simplicial presheaves.

Proposition

Every simplicial presheaf XX is a homotopy colimit over a diagram of Set-valued sheaves regarded as discrete simplicial sheaves.

More precisely, for X:S opSSetX : S^{op} \to SSet a simplicial presheaf, let D X:Δ op[S op,Set][S op,SSet]D_X : \Delta^{op} \to [S^{op},Set] \hookrightarrow [S^{op},SSet] be given by D X:[n]X nD_X : [n] \mapsto X_n. Then there is a weak equivalence

hocolim [n]ΔD X([n])X. hocolim_{[n] \in \Delta} D_X([n]) \stackrel{\simeq}{\to} X \,.
Proof

See for instance remark 2.1, p. 6

(which is otherwise about descent for simplicial presheaves).

Corollary

Let [,]:(SSet S op) op×SSet S opSSet[-,-] : (SSet^{S^{op}})^{op} \times SSet^{S^{op}} \to SSet be the canonical SSetSSet-enrichment of the category of simplicial presheaves (i.e. the assignment of SSet-enriched functor categories).

It follows in particular from the above that every such hom-object [X,A][X,A] of simplical presheaves can be written as a homotopy limit (in SSet for instance realized as a weighted limit, as described there) over evaluations of AA.

Proof

First the above yields

[X,A] [hocolim [n]ΔX n,A] holim [n]Δ[X n,A]. \begin{aligned} [X, A ] & \simeq [ hocolim_{[n] \in \Delta} X_n , A ] \\ & holim_{[n] \in \Delta} [X_n, A] \end{aligned} \,.

Next from the co-Yoneda lemma we know that the Set-valued presheaves X nX_n are in turn colimits over representables in SS, so that

holim [n]Δ[colim iU i,A] holim [n]Δlim i[U i,A]. \begin{aligned} \cdots & \simeq holim_{[n] \in \Delta} [ colim_i U_{i}, A] \\ & \simeq holim_{[n] \in \Delta} lim_i [ U_{i}, A] \end{aligned} \,.

And finally the Yoneda lemma reduces this to

holim [n]Δlim iA(U i). \begin{aligned} \cdots & holim_{[n] \in \Delta} lim_i A(U_i) \end{aligned} \,.

Notice that these kinds of computations are in particular often used when checking/computing descent and codescent along a cover or hypercover. For more on that in the context of simplicial presheaves see descent for simplicial presheaves.

Applications appear for instance at

References

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

Further articles:

  • J. F. Jardine, Stacks and the homotopy theory of simplicial sheaves. Homology, Homotopy and Applications 3:2 (2001), 361-384. doi.

  • J. F. Jardine, Fields Lectures: Simplicial presheaves.

    PDF.

For their interpretation in the more general context of (infinity,1)-sheaves see Section 6.5.2 of

Last revised on April 9, 2021 at 13:17:41. See the history of this page for a list of all contributions to it.