Simplicial presheaf
# Simplicial presheaf

In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking values in simplicial sets (i.e., a contravariant functor from the site to the category of simplicial sets). Equivalently, a simplicial presheaf is a simplicial object in the category of presheaves on a site. The notion was introduced by A. Joyal in the 1970s. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.

Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf $\operatorname {Hom} (-,U)$. Thus, a simplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf).

Example: Let G be a presheaf of groupoids. Then taking nerves section-wise, one obtains a simplicial presheaf $BG$. For example, one might set $B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}}$. These types of examples appear in K-theory.

If $f:X\to Y$ is a local weak equivalence of simplicial presheaves, then the induced map $\mathbb {Z} f:\mathbb {Z} X\to \mathbb {Z} Y$ is also a local weak equivalence.

## Homotopy sheaves of a simplicial presheaf

Let F be a simplicial presheaf on a site. The homotopy sheaves $\pi _{*}F$ of F is defined as follows. For any $f:X\to Y$ in the site and a 0-simplex s in F(X), set $(\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))$ and $(\pi _{i}^{\text{pr}}(F,s))(f)=\pi _{i}(F(Y),f^{*}(s))$. We then set $\pi _{i}F$ to be the sheaf associated with the pre-sheaf $\pi _{i}^{\text{pr}}F$.

## Model structures

The category of simplicial presheaves on a site admits many different model structures.

Some of them are obtained by viewing simplicial presheaves as functors

    $S^{op}\to \Delta ^{op}Sets$

The category of such functors is endowed with (at least) three model structures, namely the projective, the Reedy, and the injective model structure. The weak equivalences / fibrations in the first are maps

    ${\mathcal {F}}\to {\mathcal {G}}$

such that

    ${\mathcal {F}}(U)\to {\mathcal {G}}(U)$

is a weak equivalence / fibration of simplicial sets, for all U in the site S. The injective model structure is similar, but with weak equivalences and cofibrations instead.

## Stack

A simplicial presheaf F on a site is called a stack if, for any X and any hypercovering H →X, the canonical map

    $F(X)\to \operatorname {holim} F(H_{n})$

is a weak equivalence as simplicial sets, where the right is the homotopy limit of

    $[n]=\{0,1,\dots ,n\}\mapsto F(H_{n})$.

Any sheaf F on the site can be considered as a stack by viewing $F(X)$ as a constant simplicial set; this way, the category of sheaves on the site is included as a subcategory to the homotopy category of simplicial presheaves on the site. The inclusion functor has a left adjoint and that is exactly $F\mapsto \pi _{0}F$.

If A is a sheaf of abelian group (on the same site), then we define $K(A,1)$ by doing classifying space construction levelwise (the notion comes from the obstruction theory) and set $K(A,i)=K(K(A,i-1),1)$. One can show (by induction): for any X in the site,

    $\operatorname {H} ^{i}(X;A)=[X,K(A,i)]$

where the left denotes a sheaf cohomology and the right the homotopy class of maps.

