# 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.[1] Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.[2]

Example: Let us consider, say, the étale site of a scheme S. Each U in the site represents the presheaf ${\displaystyle \operatorname {Hom} (-,U)}$. 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 ${\displaystyle BG}$. For example, one might set ${\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} }$. These types of examples appear in K-theory.

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

## Homotopy sheaves of a simplicial presheaf

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

## 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

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

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

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

such that

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

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 HX, the canonical map

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

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

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

Any sheaf F on the site can be considered as a stack by viewing ${\displaystyle F(X)}$ 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 ${\displaystyle F\mapsto \pi _{0}F}$.

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

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

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