On Simplicial Principal Bundles in Descent Categories

Simplicial objects $\mathsf{sC}$ in descent categories $\mathsf{C}$, 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 $G$-bundle to the context of principal $G$-bundles in $\mathsf{sC}_{/X}$. As an application, we show how this specializes to $\mathsf{C}=\mathsf{Sets}$ to give a streamlined construction of $k$-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.