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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
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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. Similarly, a
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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
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Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study
Institute, Cambridge, UK, 9--20 September 2002
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1391:. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 131. Dordrecht: Kluwer Academic. pp. 29–68.
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Jardine, J.F. (2004). "Generalised sheaf cohomology theories". In
Greenlees, J. P. C. (ed.).
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Introductory
Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory
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where the left denotes a sheaf cohomology and the right the homotopy class of maps.
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by doing classifying space construction levelwise (the notion comes from the
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is a local weak equivalence of simplicial presheaves, then the induced map
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Some of them are obtained by viewing simplicial presheaves as functors
202:{\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} }
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The category of simplicial presheaves on a site admits many different
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is a sheaf of abelian group (on the same site), then we define
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284:{\displaystyle \mathbb {Z} f:\mathbb {Z} X\to \mathbb {Z} Y}
458:{\displaystyle (\pi _{0}^{\text{pr}}F)(X)=\pi _{0}(F(X))}
917:{\displaystyle F(X)\to \operatorname {holim} F(H_{n})}
813:{\displaystyle {\mathcal {F}}(U)\to {\mathcal {G}}(U)}
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Simplicial presheaves and derived algebraic geometry
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610:to be the sheaf associated with the pre-sheaf
209:. These types of examples appear in K-theory.
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1291:{\displaystyle \operatorname {H} ^{i}(X;A)=}
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849:on a site is called a stack if, for any
303:be a simplicial presheaf on a site. The
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1213:. One can show (by induction): for any
638:{\displaystyle \pi _{i}^{\text{pr}}F}
16:In mathematics, more specifically in
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1206:{\displaystyle K(A,i)=K(K(A,i-1),1)}
76:in the site represents the presheaf
1337:"Stacks and Non-abelian cohomology"
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1083:{\displaystyle F\mapsto \pi _{0}F}
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1316:N-group (category theory)
603:{\displaystyle \pi _{i}F}
329:{\displaystyle \pi _{*}F}
365:{\displaystyle f:X\to Y}
237:{\displaystyle f:X\to Y}
1440:J.F. Jardine's homepage
1416:"Simplicial presheaves"
1335:Toën, Bertrand (2002),
1414:Jardine, J.F. (2007).
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1125:{\displaystyle K(A,1)}
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64:Example: Consider the
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1357:Jardine 2007
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1311:cubical set
1136:) and set
765:such that
32:(e.g., the
1449:Categories
1407:1063.55004
1380:References
1020:Any sheaf
114:. Thus, a
66:étale site
1425:B. Toën,
1238:
1186:−
1069:π
1065:↦
985:↦
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893:
887:→
792:→
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688:Δ
684:→
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554:∗
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357:→
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315:π
271:→
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1465:Functors
1305:See also
853:and any
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26:presheaf
384:), set
72:. Each
59:sheaves
1405:
1395:
1346:, MSRI
127:nerves
1419:(PDF)
1340:(PDF)
1322:Notes
927:is a
890:holim
835:Stack
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1393:ISBN
1359:, §1
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