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presheaf (Definition)

For a topological space $ X$ a presheaf $ F$ with values in a category $ \mathcal{C}$ associates to each open set $ U\subset X$, an object $ F(U)$ of $ \mathcal{C}$ and to each inclusion $ U\subset V$ a morphism of $ \mathcal{C}$, $ \rho_{UV} : F(V)\to F(U)$, the restriction morphism. It is required that $ \rho_{UU} = 1_{F(U)}$ and $ \rho_{UW} = \rho_{UV}\circ \rho_{VW}$ for any $ U\subset V\subset W$.

A presheaf with values in the category of sets (or abelian groups) is called a presheaf of sets (or abelian groups). If no target category is specified, either the category of sets or abelian groups is most likely understood.

A more categorical way to state it is as follows. For $ X$ form the category $ {\bf Top}(X)$ whose objects are open sets of $ X$ and whose morphisms are the inclusions. Then a presheaf is merely a contravariant functor $ {\bf Top}(X)\to\mathcal{C}$.



"presheaf" is owned by nerdy2.
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Keywords:  topological space, category, functor
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Cross-references: contravariant functor, categorical, abelian groups, category of sets, restriction, morphism, inclusion, object, open set, associates, category, topological space
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This is version 2 of presheaf, born on 2001-12-20, modified 2002-08-24.
Object id is 1105, canonical name is Presheaf.
Accessed 2176 times total.

Classification:
AMS MSC18F20 (Category theory; homological algebra :: Categories and geometry :: Presheaves and sheaves)
 54B40 (General topology :: Basic constructions :: Presheaves and sheaves)

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