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simplicial presheaf

Context

Homotopy theory

(,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 opSetSSetD_X : \Delta^{op} \to Set \hookrightarrow 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

  • Daniel Dugger, Sharon Hollander, Daniel C. Isaksen, Hypercovers and simplicial presheaves (web)

(which is otherwise about descent for simplicial presheaves).

Corollary

Let [,]:SSet S opSSet[-,-] : 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 XX.

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 theory of simplicial presheaves and of simplicial sheaves was developed by J. 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 Toë as described at infinity-stack homotopically.

  • JardStackSSh – J. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homology, homotopy and applications, vol. 3(2), 2001 p. 361-284 (pdf)
  • JardSimpSh – J. Jardine, Fields Lectures: Simplicial presheaves (pdf)

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

Revised on September 24, 2013 15:38:32 by Tim Porter (95.147.236.191)