# Simplicial presheaf

In mathematics, more specifically inhomotopy theory, asimplicial presheafis apresheafon a站点(e.g., the多类别属于topological spaces) taking values insimplicial sets(i.e., acontravariant functorfrom 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, asimplicial sheafon a site is asimplicial objectin the category ofsheaveson the site.[2]

Example: Let us consider, say, theétale siteof a schemeSEachUin the site represents the presheaf${\displaystyle \operatorname {Hom} (-,U)}$Thus, asimplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf).

Example: LetGbe a presheaf of groupoids. Then taking神经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

LetFbe a simplicial presheaf on a site. 这个homotopy sheaves ${\displaystyle \pi _{*}F}$属于Fis defined as follows. For any${\displaystyle f:X\to Y}$in the site and a 0-simplexs进入F(X), set${\displaystyle (\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))}$${\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 differentmodel 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 presheafFon a site is called a stack if, for anyXand anyhypercovering HX, the canonical map

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

is aweak equivalenceas simplicial sets, where the right is thehomotopy limit属于

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

Any sheafFon 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${\displaystyle F\mapsto \pi _{0}F}$

Ifis 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 theobstruction theory) and set${\displaystyle K(A,i)=K(K(A,i-1),1)}$One can show (by induction): for anyXin 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.