Pre-Sheaf

A pre-sheaf is a contravariant set-valued functor. That is to say, a pre-sheaf \(F\) on a category \(C\) is a functor \(F: C^{\text{op}} \to \text{Set}\).

Sheaves

Let \(X\) be a set and \(F: \mathcal{O}(X)^{\text{op}} \to \text{Set}\) be a pre-sheaf on the poset category of \(X\), \(\mathcal{O}(X)\). \(F\) is a sheaf if it preserves all covering colimits.