nLab presheaf on open subsets

The notion of presheaf today is technically synonymous with (contravariant) functor $F : S^{op} \to A$ – albeit a functor is called a presheaf only if it is being used in a certain way, notably when its domain has the structure of a site and there is the potential to consider the sheafification of the functor to a sheaf.

Historically this formalization was abstracted from the special case where the domain category $S = O(X)$ is the category of open subsets of a topological space.

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