Simplicial presheaf

 ( Homotopy Theory ) 

In mathematics, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the Category theory of topological spaces) taking values in (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. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.

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Examples Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf $\backslash operatorname\{Hom\}(-,\; U)$. 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 . Then taking nerves section-wise, one obtains a simplicial presheaf $BG$. For example, one might set $B\backslash operatorname\{GL\}\; =\; \backslash varinjlim\; B\backslash operatorname\{GL\_n\}$. These types of examples appear in K-theory.

If $f:\; X\; \backslash to\; Y$ is a local weak equivalence of simplicial presheaves, then the induced map $\backslash mathbb\{Z\}\; f:\; \backslash mathbb\{Z\}\; X\; \backslash to\; \backslash mathbb\{Z\}\; Y$ is also a local weak equivalence.

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Homotopy sheaves of a simplicial presheaf Let F be a simplicial presheaf on a site. The homotopy sheaves $\backslash pi\_*\; F$ of F are defined as follows. For any $f:X\; \backslash to\; Y$ in the site and a 0-simplex s in F( X), set $(\backslash pi\_0^\backslash text\{pr\}\; F)(X)\; =\; \backslash pi\_0\; (F(X))$ and $(\backslash pi\_i^\backslash text\{pr\}\; (F,\; s))(f)\; =\; \backslash pi\_i\; (F(Y),\; f^*(s))$. We then set $\backslash pi\_i\; F$ to be the sheaf associated with the pre-sheaf $\backslash pi\_i^\backslash text\{pr\}\; F$.

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

Some of them are obtained by viewing simplicial presheaves as functors

$S^\{op\}\; \backslash to\; \backslash Delta^\{op\}\; Sets$

The category of such functors is endowed with (at least) three model structures, namely the projective, the Reedy category, and the injective model structure. The weak equivalences / fibrations in the first are maps

$\backslash mathcal\; F\; \backslash to\; \backslash mathcal\; G$

such that

$\backslash mathcal\; F(U)\; \backslash to\; \backslash mathcal\; G(U)$

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.

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Stack A simplicial presheaf F on a site is called a stack if, for any X and any hypercovering H → X, the canonical map

$F(X)\; \backslash to\; \backslash operatorname\{holim\}\; F(H\_n)$

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

$n\; =\; \backslash \{\; 0,\; 1,\; \backslash dots,\; n\; \backslash \}\; \backslash mapsto\; F(H\_n)$.

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 $F\; \backslash mapsto\; \backslash pi\_0\; F$.

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,

$\backslash operatorname\{H\}^i(X;\; A)\; =\; X,$

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

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See also

• cubical set
• N-group (category theory)

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Notes

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Further reading

• Konrad Voelkel, Model structures on simplicial presheaves

• J.F. Jardine (2026). 9781402018336, Kluwer Academic. ISBN 9781402018336
• B. Toën, Simplicial presheaves and derived algebraic geometry

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External links

• J.F. Jardine's homepage

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