sheaf

1 Presheaves

Let $X$ be a topological space and let $𝒜$ be a category. A presheaf on $X$ with values in $𝒜$ is a contravariant functor $F$ from the category $ℬ$ whose objects are open sets in $X$ and whose morphisms 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 equivalent (but longer) definition. A presheaf $F$ on $X$ consists of the following data:

1. An object $F(U)$ in $𝒜$, for each open set $U\subset X$

2. A morphism ${\mathrm{res}}_{V,U}:F(V)\to F(U)$ for each pair of open sets $U\subset V$ in $X$ (called the restriction morphism), such that:

   1. For every open set $U\subset X$, the morphism ${\mathrm{res}}_{U,U}$ is the identity morphism.

   2. For any open sets $U\subset V\subset W$ in $X$, the diagram