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]

## Contents

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

## Examples

Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf 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 BGL⁡=lim->BGLn. These types of examples appear in K-theory.

If f:X->Y is a local weak equivalence of simplicial presheaves, then the induced map ℤf:ℤX->ℤY is also a local weak equivalence.

## Homotopy sheaves of a simplicial presheaf

Let F be a simplicial presheaf on a site. The homotopy sheaves π*F of F are defined as follows. For any f:X->Y in the site and a 0-simplex s in F(X), set (π0prF)(X)=π0(F(X)) and (πipr(F,s))(f)=πi(F(Y),f*(s)). We then set πiF to be the sheaf associated with the pre-sheaf πiprF.

## Model structures

The category of simplicial presheaves on a site admits several different model structures.

Some of them are obtained by viewing simplicial presheaves as functors

Sop->ΔopSets

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

ℱ->𝒢

such that

ℱ(U)->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)->holim⁡F(Hn)

is a weak equivalence as simplicial sets, where the right is the homotopy limit of

[n]={0,1,…,n}↦F(Hn).

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↦π0F.

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,

Hi(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)

## Notes

1. ↑ Toën, Bertrand (2002), "Stacks and Non-abelian cohomology", Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory, MSRI, https://perso.math.univ-toulouse.fr/btoen/files/2015/02/msri2002.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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