Simplicial presheaf

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In mathematics, more specifically inhomotopy theory, asimplicial presheafis apresheafon a站点(e.g., the多类别属于topological spaces) taking values insimplicial sets(i.e., acontravariant functorfrom 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, asimplicial sheafon a site is asimplicial objectin the category ofsheaveson the site.[2]

Example: Let us consider, say, theétale siteof a schemeSEachUin the site represents the presheafThus, asimplicial scheme, a simplicial object in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf).

Example: LetGbe a presheaf of groupoids. Then taking神经section-wise, one obtains a simplicial presheafFor example, one might setThese types of examples appear in K-theory.

Ifis a local weak equivalence of simplicial presheaves, then the induced mapis also a local weak equivalence.

Homotopy sheaves of a simplicial presheaf[编辑]

LetFbe a simplicial presheaf on a site. 这个homotopy sheaves 属于Fis defined as follows. For anyin the site and a 0-simplexs进入F(X), setWe then setto be the sheaf associated with the pre-sheaf

Model structures[编辑]

The category of simplicial presheaves on a site admits many differentmodel structures

Some of them are obtained by viewing simplicial presheaves as functors

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

such that

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.


A simplicial presheafFon a site is called a stack if, for anyXand anyhypercovering HX, the canonical map

is aweak equivalenceas simplicial sets, where the right is thehomotopy limit属于

Any sheafFon the site can be considered as a stack by viewingas 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

Ifis a sheaf of abelian group (on the same site), then we defineby doing classifying space construction levelwise (the notion comes from theobstruction theory) and setOne can show (by induction): for anyXin the site,

where the left denotes a sheaf cohomology and the right the homotopy class of maps.

See also[编辑]


Further reading[编辑]


  • 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 2002NATO Science Series II: Mathematics, Physics and Chemistry.一百三十一Dordrecht: Kluwer Academic. pp. 29–68.ISBN 1-4020-1833-9Zbl 一千零六十三点五五零零四
  • Jardine, J.F. (2007)."Simplicial presheaves" (PDF)
  • B. Toën,Simplicial presheaves and derived algebraic geometry

External links[编辑]