Simplicial presheaf explained

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.

Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf

\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

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 Z f: Z X \to 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

\pi _{*} F of F is defined as follows. For any f:X \to Y in the site and a 0-simplex s in F(X), set (\pi ^{pr} _{0} F)(X) = \pi _{0} (F(X)) and (\pi ^{pr} _{i} (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 ^{pr} _{i} 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

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 mapsl F \to l G such that l F(U) \to l 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

See main article: Stack (mathematics).

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) \to \operatorname{holim} F(H _{n)} is a weak equivalence as simplicial sets, where the right is the homotopy limit of[n] = \{ 0 , 1 , ... , n \} \mapsto F(H _{n)} .

Any sheaf F on 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 A 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 the obstruction theory) and set K(A, i) = K(K(A, i -1 ), 1 ) . One can show (by induction): for any X in 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.

See also

• cubical set
• N-group (category theory)

Further reading

• Konrad Voelkel, Model structures on simplicial presheaves

References

• Book: Jardine, J.F. . Generalised sheaf cohomology theories . 29–68 . Greenlees . J. P. C. . Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, 9--20 September 2002 . Dordrecht . Kluwer Academic . NATO Science Series II: Mathematics, Physics and Chemistry . 131 . 2004 . 1-4020-1833-9 . 1063.55004 .
• Web site: J.F. . Jardine . 2007 . Simplicial presheaves .
• B. Toën, Simplicial presheaves and derived algebraic geometry

External links

• J.F. Jardine's homepage