nLab simplicial presheaf

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.

Remarks

• There are various useful model category structures on the category of simplicial presheaves. See model structure on simplicial presheaves.

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

• Daniel Dugger, Sharon Hollander, Daniel Isaksen, Hypercovers and simplicial presheaves, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 136 Issue 1, 2004 (arXiv:math/0205027, K-theory:0563, doi:10.1017/S0305004103007175)

(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.

Related entries

• model structure on simplicial presheaves

• descent for simplicial presheaves

• presheaf of spectra

Applications appear for instance at

• geometric infinity-function theory

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.

The Joyal model structure on simplicial sheaves was constructed in 1984 by Joyal in

• André Joyal, Lettre d’André Joyal à Alexandre Grothendieck, April 11, 1984, PDF.

Simplicial sheaves were endowed with a structure of a category of fibrant objects in

• 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

The model structure on simplicial presheaves is due to Jardine:

• J. F. Jardine, Simplical presheaves. Journal of Pure and Applied Algebra 47:1 (1987), 35-87. doi

A modern expository account is in

• John F. Jardine, Local Homotopy Theory, Springer, 2015. doi.

The local projective model structure (as opposed to the injective model structures used above) on simplicial presheaves and simplicial sheaves is due to

• Benjamin A. Blander?, Local Projective Model Structures on Simplicial Presheaves, K-Theory 24:3, 283-301. doi.

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

• Jacob Lurie, Higher Topos Theory.