Cocycle categories

Suppose that G is a sheaf of groups on a space X and that Uα ⊂ X is an open covering. Then a cocycle for the covering is traditionally defined to be a family of elements gαβ ∈ G(Uα ∩ Uβ) such that gαα = e and gαβgβγ = gαγ when all elements are restricted to the group G(Uα ∩ Uβ ∩ Uγ). A more compact way of saying this is to assert that such a cocycle is a map of simplicial sheaves C(U)→ BG on the space X , where C(U) is the Čech resolution associated to the covering family {Uα}. The canonical map C(U)→ ∗ is a local weak equivalence of simplicial sheaves, and is a fibration in each section since C(U) is actually the nerve of a groupoid — the map C(U)→ ∗ is therefore a hypercover, which is most properly defined to be a map of simplicial sheaves (or presheaves) which is a Kan fibration and a weak equivalence in each stalk. Every cocycle in the traditional sense therefore determines a picture of simplicial sheaf morphisms ∗ ← C(U)→ BG.