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 ${\displaystyle \pi _{i}^{\text{pr}}F}$.

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.

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

• cubical set
• N-group (category theory)

• Konrad Voelkel, Model structures on simplicial presheaves

• Jardine, J.F. (2004). "Generalised sheaf cohomology theories". In Greenlees, J. P. C. (ed.). 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. Vol. 131. Dordrecht: Kluwer Academic. pp. 29–68. ISBN 1-4020-1833-9. Zbl 1063.55004.
• Jardine, J.F. (2007). "Simplicial presheaves" (PDF).
• B. Toën, Simplicial presheaves and derived algebraic geometry

• J.F. Jardine's homepage