sheaf

Let be a topological space^{👁 Mathworld
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and let be a category^{👁 Mathworld
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. A presheaf^{👁 Planetmath
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on with values in is a contravariant functor^{👁 Mathworld
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from the category whose objects are open sets in and whose morphisms^{👁 Mathworld
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are inclusion mappings of open sets of , to the category .

As this definition may be less than helpful to many readers, we offer the following equivalent^{👁 Mathworld
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(but longer) definition. A presheaf on consists of the following data:

1. 1.

   An object in , for each open set

2. 2.

   A morphism for each pair of open sets in (called the restriction^{👁 Planetmath
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   morphism), such that:

   1. (a)

      For every open set , the morphism is the identity morphism.

   2. (b)

      For any open sets in , the diagram