Cocycles in categories of fibrant objects

TL;DR

This paper demonstrates that categories of fibrant objects support a homotopical calculus of right fractions via cocycles, enabling non-abelian versions of the Verdier hypercovering theorem.

Contribution

It introduces a homotopical calculus of cocycles for categories of fibrant objects, extending the Dwyer-Kan calculus to this setting.

Findings

Categories of fibrant objects admit a homotopical calculus of right fractions.

The calculus of cocycles can be constructed on any category of fibrant objects.

Derived non-abelian versions of the Verdier hypercovering theorem.

Abstract

We establish that a category of fibrant objects (in the sense of Brown) admits a Dwyer-Kan homotopical calculus of right fractions. This is done using a homotopical calculus of cocycles, which is an auxiliary structure that can be defined on every category of fibrant objects. As an application, we deduce some non-abelian versions of the Verdier hypercovering theorem.