Cocycles

Abstract

The category of cocycles between a pair of objects in a model category is introduced. Examples of cocycles include the usual geometric constructions in topological and algebraic settings, but the theory is much more general.

The primary theorem, that one can recover morphisms in the homotopy category from path components in cocycle categories, applies to a large selection of model structures. This result is one of the most useful formal ideas in local homotopy theory: it is used in later chapters to establish homotopy classification results for abelian and non-abelian sheaf cohomology theories, and to establish a theory of cup products. The theory of cocycles also leads to a fast proof of a widely applicable version of the Verdier hypercovering theorem.