For X a topological space, the presheaf ℱ of Abelian groups (rings, ...) on X is defined such that 1. For every open subset U⊆X, an Abelian group (ring, ...) ℱ(U), and 2. For every inclusion V⊆U of open subsets of X, a morphism of Abelian groups (rings, ...) ρ_(U V) :ℱ(U)->ℱ(V) subject to the conditions: 1. If ∅ denotes the empty set, then ℱ(∅) = 0, 2.ρ_(U U) is the identity map ℱ(U)->ℱ(U), and 3. If W⊆V⊆U are three open subsets, then ρ_(U W) = ρ_(V W) °ρ_(U V).

presheaf of categories