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: Consider the étale site of a scheme S. Each U in the site represents the presheaf [math]\displaystyle{ \operatorname{Hom}(-, U) }[/math]. 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 [math]\displaystyle{ BG }[/math]. For example, one might set [math]\displaystyle{ B\operatorname{GL} = \varinjlim B\operatorname{GL_n} }[/math]. These types of examples appear in K-theory.
If [math]\displaystyle{ f: X \to Y }[/math] is a local weak equivalence of simplicial presheaves, then the induced map [math]\displaystyle{ \mathbb{Z} f: \mathbb{Z} X \to \mathbb{Z} Y }[/math] is also a local weak equivalence.
Homotopy sheaves of a simplicial presheaf
Let F be a simplicial presheaf on a site. The homotopy sheaves [math]\displaystyle{ \pi_* F }[/math] of F is defined as follows. For any [math]\displaystyle{ f:X \to Y }[/math] in the site and a 0-simplex s in F(X), set [math]\displaystyle{ (\pi_0^\text{pr} F)(X) = \pi_0 (F(X)) }[/math] and [math]\displaystyle{ (\pi_i^\text{pr} (F, s))(f) = \pi_i (F(Y), f^*(s)) }[/math]. We then set [math]\displaystyle{ \pi_i F }[/math] to be the sheaf associated with the pre-sheaf [math]\displaystyle{ \pi_i^\text{pr} F }[/math].
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
- [math]\displaystyle{ S^{op} \to \Delta^{op} Sets }[/math]
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
- [math]\displaystyle{ \mathcal F \to \mathcal G }[/math]
such that
- [math]\displaystyle{ \mathcal F(U) \to \mathcal G(U) }[/math]
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
- [math]\displaystyle{ F(X) \to \operatorname{holim} F(H_n) }[/math]
is a weak equivalence as simplicial sets, where the right is the homotopy limit of
- [math]\displaystyle{ [n] = \{ 0, 1, \dots, n \} \mapsto F(H_n) }[/math].
Any sheaf F on the site can be considered as a stack by viewing [math]\displaystyle{ F(X) }[/math] 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 [math]\displaystyle{ F \mapsto \pi_0 F }[/math].
If A is a sheaf of abelian group (on the same site), then we define [math]\displaystyle{ K(A, 1) }[/math] by doing classifying space construction levelwise (the notion comes from the obstruction theory) and set [math]\displaystyle{ K(A, i) = K(K(A, i-1), 1) }[/math]. One can show (by induction): for any X in the site,
- [math]\displaystyle{ \operatorname{H}^i(X; A) = [X, K(A, i)] }[/math]
where the left denotes a sheaf cohomology and the right the homotopy class of maps.
See also
Notes
Further reading
- Konrad Voelkel, Model structures on simplicial presheaves
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.
- Jardine, J.F. (2007). "Simplicial presheaves". http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf.
- B. Toën, Simplicial presheaves and derived algebraic geometry
External links
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