Sheaf (mathematics)

In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves, it turns out, enable one to discuss in a refined way what is a local property, as applied to a function.

Morphisms of sheaves

Let F and G be two sheaves on X both with values in the category C. We define a morphism from G to F to be a family of morphisms φU : G(U) → F(U) in the category C for all opens U in X which commute with the restriction maps. That is, the following diagram must commute

Related Topics:
Morphism - Commute

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for each pair of open sets U ⊆ V in X. If F and G are considered as contravariant functors from TopX to C then a morphism between them is nothing more than a natural transformation. With this definition the set of all C-valued sheaves on X forms a category (a functor category). An isomorphism of sheaves on X is just an isomorphism in this category.

Related Topics:
Contravariant functor - Natural transformation - Functor category

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One can generalize this notion to morphisms between sheaves on different spaces. Let f : X → Y be a continuous function between two topological spaces, and let F be a sheaf on X and G a sheaf on Y both with values in C. Then a morphism from G to F relative to f is given by a family of morphisms φU : G(U) → F(f−1(U)) for each open set U in Y such that the diagram

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commutes for each pair of open sets U ⊆ V in Y. The previous definition is the special case resulting when f is the identity map on X.

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The category theoretical description is slightly more complicated in the general case. Let Top be the contravariant functor from the category of topological spaces Top to the category of small categories Cat which sends each space X to the poset category of its open sets TopX. Here Top(f) is a covariant functor from TopY to TopX sending each open set to its preimage. Composing F with Top(f) we obtain a contravariant functor from TopY to C. A morphism from G to F relative to f is then a natural transformation from G to F ○ Top(f).

Related Topics:
Category of topological spaces - Category of small categories - Preimage

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Note that all of the above makes sense if we are working only with presheaves instead of sheaves.

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