The Verdier hypercovering theorem

to the morphism f ·p−1, is a bijection provided that X and Y are locally fibrant in the sense that all of their stalks are Kan complexes. Here, π(Z, Y ) denotes simplicial homotopy classes of maps, and the colimit is indexed over homotopy classes represented by hypercovers p : Z → X. A hypercover is a map which is a trivial fibration of simplicial sets in all stalks, and is therefore invertible in the homotopy category. The theorem is stated in the form displayed above (for simplicial sheaves) in [7], and the proof given there is a calculus of fractions argument which is adapted from Brown’s thesis [3]. A more recent version of the classical proof, for simplicial presheaves, appears in [4]. The Verdier hypercovering theorem had multiple applications throughout the early development of simplicial sheaf homotopy theory, such as the identification of sheaf cohomology with homotopy classes of maps which appeared in [7] and [9] for abelian and non-abelian cohomology, respectively. Variants of the theorem have become part of the basic tool kit for all work in this type of homotopy theory — see [6], for example.