sheaf
1 Presheaves
Let be a topological space and let be a category. A presheaf on with values in is a contravariant functor from the category whose objects are open sets in and whose morphisms 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 (but longer) definition. A presheaf on consists of the following data:
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1.
An object in , for each open set
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2.
A morphism for each pair of open sets in (called the restriction morphism), such that:
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(a)
For every open set , the morphism is the identity morphism.
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(b)
For any open sets in , the diagram
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(a)