Simplicial presheaf

In mathematics, more specifically in <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a>, a simplicial presheaf is a <a href="/facts/Presheaf/MM3hcRw4">presheaf</a> on a <a href="/facts/Site_(mathematics)/Przxx5sH">site</a> (e.g., the <a href="/facts/Category_theory/6zS3DXPa">category</a> of <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a>) taking values in <a href="/facts/Simplicial_set/6DaqXMLT">simplicial sets</a> (i.e., a <a href="/facts/Contravariant_functor/l2u3rIBE">contravariant functor</a> 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 <a href="/facts/Simplicial_object/6DaqXMLT">simplicial object</a> in the category of <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaves</a> on the site.
Example: Consider the <a href="/facts/%C3%89tale_site/Przxx5sH">étale site</a> of a scheme S. Each U in the site represents the presheaf 
 
 
 
 Hom
 ⁡
 (
 −
 ,
 U
 )
 
 
 {\displaystyle \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 groupoids. Then taking <a href="/facts/Nerve_(category_theory)/VceKkFGP">nerves</a> section-wise, one obtains a simplicial presheaf 
 
 
 
 B
 G
 
 
 {\displaystyle BG}
 
. For example, one might set 
 
 
 
 B
 GL
 =
 
 
 lim
 →
 
 
 ⁡
 B
 
 G
 
 L
 
 n
 
 
 
 
 
 {\displaystyle B\operatorname {GL} =\varinjlim B\operatorname {GL_{n}} }
 
. These types of examples appear in K-theory.
If 
 
 
 
 f
 :
 X
 →
 Y
 
 
 {\displaystyle f:X\to Y}
 
 is a local weak equivalence of simplicial presheaves, then the induced map 
 
 
 
 
 Z
 
 f
 :
 
 Z
 
 X
 →
 
 Z
 
 Y
 
 
 {\displaystyle \mathbb {Z} f:\mathbb {Z} X\to \mathbb {Z} Y}
 
 is also a local weak equivalence.

Simplicial presheaf open-in-new

Simplicial presheaf