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.

## Contents

* 1 Homotopy sheaves of a simplicial presheaf
* 2 Model structures
* 3 Stack
* 4 See also
* 5 Notes
* 6 Further reading
* 7 References
* 8 External links

## 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

Main page: 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

[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

* cubical set
* N-group (category theory)

## Notes

1. ↑ http://ncatlab.org/nlab/files/ToenStacksNAC.pdf[bare URL PDF]
2. ↑ Jardine 2007, §1

## 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

* J.F. Jardine's homepage

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