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Statements

Subject Item
dbr:Simplicial_presheaf
rdf:type
yago:WikicatSimplicialSets yago:Abstraction100002137 yago:Set107996689 yago:Group100031264 yago:Collection107951464
rdfs:label
Simplicial presheaf
rdfs:comment
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. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site.
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dbc:Functors dbc:Simplicial_sets dbc:Homotopy_theory
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dbr:Category_theory dbr:Weak_equivalence_(homotopy_theory) dbr:Topological_space dbr:Presheaf dbr:Contravariant_functor dbr:Site_(mathematics) dbr:Obstruction_theory dbr:Étale_site dbr:Homotopy_theory dbr:Simplicial_object dbr:Model_category dbc:Homotopy_theory dbr:Cubical_set dbc:Simplicial_sets dbr:Sheaf_(mathematics) dbr:Simplicial_set dbc:Functors dbr:Nerve_(category_theory) dbr:Hypercovering dbr:N-group_(category_theory) dbr:Homotopy_limit
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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. Similarly, a simplicial sheaf on a site is a simplicial object in the category of sheaves on the site. Example: Consider the étale site of a scheme S. Each U in the site represents the presheaf . 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 . For example, one might set . These types of examples appear in K-theory. If is a local weak equivalence of simplicial presheaves, then the induced map is also a local weak equivalence.
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wikipedia-en:Simplicial_presheaf?oldid=1076962355&ns=0
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wikipedia-en:Simplicial_presheaf