On simplicial principal bundles in descent categories

Simplicial objects s C \mathsf {sC} in descent categories C \mathsf {C} , as introduced by Behrend and Getzler, provide a context in which to study higher stacks. In this note, we extend the construction of the canonical cocycle of a smooth principal G G -bundle to the context of principal G G -bundles in s C / X \mathsf {sC}_{/X} . As an application, we show how this specializes to C = S e t s \mathsf {C}=\mathsf {Sets} to give a streamlined construction of k k -invariants of reduced Kan complexes. We adapt this to give a similarly streamlined construction of minimal Kan complexes, with the goal of clarifying the role of the axiom of choice; Postnikov towers of minimal Kan complexes provide examples of towers of simplicial principal bundles of the type we consider.