In category theory, a presheaf is a structure that assigns data to the open sets of a topological space (or more generally, to objects in a category) in a way that respects the relationships between these sets (or objects). More formally, a presheaf can be defined as follows: ### Definition: Let \( C \) be a category and \( X \) a topological space (or a more abstract site).
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A presheaf on a category is a functor . A morphism induces the restriction map , reversing direction. The indexing category need not be small. For the category of open subsets of a topological space, this specializes to a presheaf of sets on a topological space.