# nLab simplicial presheaf

Contents

### Context

#### $(\infty,1)$-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 $S^{op} \to \Simp\Set$, i.e., functors $S^{op} \to [\Delta^{op}, \Set]$;

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

## Interpretation as $\infty$-stacks

Regarding $\Simp\Set$ as a model category using the standard model structure on simplicial sets and inducing from that a model structure on $[S^{op}, \Simp\Set]$ 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 $S$ 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 $\mathbf{B}G$ and then taking the nerve $N(\mathbf{B}G) \in \Simp\Set$ of these (the “classifying simplicial set of the group whose geometric realization is the classifying space $\mathcal{B}G$), 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 $X : S^{op} \to SSet$ a simplicial presheaf, let $D_X : \Delta^{op} \to [S^{op},Set] \hookrightarrow [S^{op},SSet]$ be given by $D_X : [n] \mapsto X_n$. Then there is a weak equivalence

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

See for instance remark 2.1, p. 6

(which is otherwise about descent for simplicial presheaves).

###### Corollary

Let $[-,-] : (SSet^{S^{op}})^{op} \times SSet^{S^{op}} \to SSet$ be the canonical $SSet$-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 $[X,A]$ 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 $A$.

###### Proof

First the above yields

\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{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{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 July 1, 2021 at 10:31:25. See the history of this page for a list of all contributions to it.