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 schemeS。EachUin the site represents the presheaf$\operatorname {Hom} (-,U)$。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$BG$。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.

LetFbe a simplicial presheaf on a site. 这个homotopy sheaves$\pi _{*}F$属于Fis defined as follows. For any$f:X\to Y$in the site and a 0-simplexs进入F(X), set$(\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))$和$(\pi _{i}^{\text{pr}}(F,s))(f)=\pi _{i}(F(Y),f^{*}(s))$。We then set$\pi _{i}F$to be the sheaf associated with the pre-sheaf$\pi _{i}^{\text{pr}}F$。

The category of simplicial presheaves on a site admits many differentmodel 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.

Any sheafFon 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一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 theobstruction theory) and set$K(A,i)=K(K(A,i-1),1)$。One can show (by induction): for anyXin 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.

Jardine, J.F. (2004). "Generalised sheaf cohomology theories". In Greenlees, J. P. C.Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, 9--20 September 2002。NATO Science Series II: Mathematics, Physics and Chemistry.一百三十一。Dordrecht: Kluwer Academic. pp. 29–68.ISBN1-4020-1833-9。Zbl一千零六十三点五五零零四。