Simplicial presheaf

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.^[1] Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.^[2]

Example: Let us consider, say, 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 $\mathbb{Z} f: \mathbb{Z} X \to \mathbb{Z} Y$ is also a local weak equivalence.

Homotopy sheaves of a simplicial presheaf

Let F be a simplicial presheaf on a site. The homotopy sheaves $\pi_* F$ of F is defined as follows. For any $f:X \to Y$ in the site and a 0-simplex s in F(X), set $(\pi_0^\text{pr} F)(X) = \pi_0 (F(X))$ and $(\pi_i^\text{pr} (F, s))(f) = \pi_i (F(Y), f^*(s))$ . We then set $\pi_i F$ to be the sheaf associated with the pre-sheaf $\pi_i^\text{pr} F$ .

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 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.

Stack

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, \dots, 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.

Notes

References

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 131. Dordrecht: Kluwer Academic. pp. 29–68. ISBN 1-4020-1833-9. Zbl 1063.55004.
Jardine, J.F. (2007). "Simplicial presheaves".
B. Toën, Simplicial presheaves and derived algebraic geometry

External links

J.F. Jardine's homepage