1 Presheaves

Let X be a topological spaceMathworldPlanetmath and let 𝒜 be a categoryMathworldPlanetmath. A presheafPlanetmathPlanetmathPlanetmath on X with values in 𝒜 is a contravariant functorMathworldPlanetmath F from the category whose objects are open sets in X and whose morphismsMathworldPlanetmath are inclusion mappings of open sets of X, to the category 𝒜.

As this definition may be less than helpful to many readers, we offer the following equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (but longer) definition. A presheaf F on X consists of the following data:

  1. 1.

    An object F(U) in 𝒜, for each open set UX

  2. 2.

    A morphism resV,U:F(V)F(U) for each pair of open sets UV in X (called the restrictionPlanetmathPlanetmathPlanetmath morphism), such that:

    1. (a)

      For every open set UX, the morphism resU,U is the identity morphism.

    2. (b)

      For any open sets UVW in X, the diagram