# nLab simplicial presheaf

## Theorems

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Definition

Simplicial presheaves over some site $S$ are

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

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 ${\Delta }^{\mathrm{op}}\to \left[{S}^{\mathrm{op}},Set\right]$.

## Interpretation as $\infty$-stacks

Regarding $SimpSet$ as a model category using the standard model structure on simplicial sets and inducing from that a model structure on $\left[{S}^{\mathrm{op}},SimpSet\right]$ 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 $S$ can be regarded as an $\infty$-groupoid (in particular a Kan complex) whose space of $n$-morphisms is modeled on the objects of $S$ 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 $G$ as a one object category $BG$ and then taking the nerve $N\left(BG\right)\in SimpSet$ of these (the “classifying simplicial set of the group whose geometric realization is the classifying space $ℬG$), which is clearly a functorial operation, turns any presheaf with values in groups into a simplicial presheaf.

## Properties

Here are some basic but useful facts about simplicial presheaves.

###### Proposition

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

More precisely, for $X:{S}^{\mathrm{op}}\to \mathrm{SSet}$ a simplicial presheaf, let ${D}_{X}:{\Delta }^{\mathrm{op}}\to \mathrm{Set}↪\mathrm{SSet}$ be given by ${D}_{X}:\left[n\right]↦{X}_{n}$. Then there is a weak equivalence

${\mathrm{hocolim}}_{\left[n\right]\in \Delta }{D}_{X}\left(\left[n\right]\right)\stackrel{\simeq }{\to }X\phantom{\rule{thinmathspace}{0ex}}.$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 $\left[-,-\right]:{\mathrm{SSet}}^{{S}^{\mathrm{op}}}\to \mathrm{SSet}$ be the canonical $\mathrm{SSet}$-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 $\left[X,A\right]$ 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 $X$.

###### Proof

First the above yields

$\begin{array}{rl}\left[X,A\right]& \simeq \left[{\mathrm{hocolim}}_{\left[n\right]\in \Delta }{X}_{n},A\right]\\ & {\mathrm{holim}}_{\left[n\right]\in \Delta }\left[{X}_{n},A\right]\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${X}_{n}$ are in turn colimits over representables in $S$, so that

$\begin{array}{rl}\cdots & \simeq {\mathrm{holim}}_{\left[n\right]\in \Delta }\left[{\mathrm{colim}}_{i}{U}_{i},A\right]\\ & \simeq {\mathrm{holim}}_{\left[n\right]\in \Delta }{\mathrm{lim}}_{i}\left[{U}_{i},A\right]\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{rl}\cdots & {\mathrm{holim}}_{\left[n\right]\in \Delta }{\mathrm{lim}}_{i}A\left({U}_{i}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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)