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.
Examples Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf Hom ( − , U ) {\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 B G {\displaystyle BG} . For example, one might set B GL = lim → B G L n {\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} } . These types of examples appear in K-theory. If f : X → Y {\displaystyle f:X\to Y} is a local weak equivalence of simplicial presheaves, then the induced map Z f : Z X → Z Y {\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 π ∗ F {\displaystyle \pi _{*}F} of F are defined as follows. For any f : X → Y {\displaystyle f:X\to Y} in the site and a 0-simplex s in F(X), set ( π 0 pr F ) ( X ) = π 0 ( F ( X ) ) {\displaystyle (\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))} and ( π i pr ( F , s ) ) ( f ) = π i ( F ( Y ) , f ∗ ( s ) ) {\displaystyle (\pi _{i}^{\text{pr}}(F,s))(f)=\pi _{i}(F(Y),f^{*}(s))} . We then set π i F {\displaystyle \pi _{i}F} to be the sheaf associated with the pre-sheaf π i pr F {\displaystyle \pi _{i}^{\text{pr}}F} .
Model structures The category of simplicial presheaves on a site admits several different model structures. Some of them are obtained by viewing simplicial presheaves as functors
S o p → Δ o p S e t s {\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
F → G {\displaystyle {\mathcal {F}}\to {\mathcal {G}}}
such that
F ( U ) → G ( U ) {\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 H →X, the canonical map
F ( X ) → holim F ( H n ) {\displaystyle F(X)\to \operatorname {holim} F(H_{n})}
is a weak equivalence as simplicial sets, where the right is the homotopy limit of
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