PlanetMath: presheaf of a topological basis

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presheaf of a topological basis

(Definition)

Let be a topological space and let $\mathcal B$ be a basis of its topology. We can regard $\mathcal B$ as a category with objects being the open sets in $\mathcal B$ and arrows/morphisms between $U,V\in\mathcal B$ to exists only if $U\subset V$ , and where the only element of $\mathcal B(U,V)$ is the injection map $U\hookrightarrow V$ . Let now $\Ccal$ be a complete category, we now define the presheaf of $\Ccal$ -objects over the basis $\mathcal B$ of the topology of to be a contravariant functor

$\displaystyle \P:\mathcal B\rightarrow \Ccal$

"presheaf of a topological basis" is owned by jocaps.

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Cross-references: contravariant functor, presheaf, complete category, map, injection, open sets, objects, category, basis, topological space
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This is version 11 of presheaf of a topological basis, born on 2006-11-04, modified 2006-11-04.
Object id is 8519, canonical name is SomethingRelatedToSheaf.
Accessed 528 times total.

Classification:

AMS MSC:	18F20 (Category theory; homological algebra :: Categories and geometry :: Presheaves and sheaves)
	54B40 (General topology :: Basic constructions :: Presheaves and sheaves)
	14F05 (Algebraic geometry :: homology theory :: Vector bundles, sheaves, related constructions)

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