Let $M$ be a smooth manifold, let $G$ be a Lie group, and fix a numerable open cover $\mathcal U=(U_i)_{i\in I}$ of $M$. Smooth principal right $G$-bundles over $M$ that are trivializable over this cover, considered up to principal bundle isomorphism over $M$, are classified by coboundary equivalence classes of smooth $G$-valued nonabelian Cech $1$-cocycles:

\begin{align*} \{g_{ij}:U_i\cap U_j\to G\mid g_{ij}g_{jk}=g_{ik}\}/\sim. \end{align*}