A principle bundle over a topological space can be given by transitions functions. On the other hand, in the simplicial world principle bundles can be given by twisting functions. Is there an explicit description of this relationship written up somewhere? Or can anyone explain how it can be obtained?
Given a good topological space $B$ and a topological group $G$, a $G$-principle bundle over $B$ can be given by a collection of transition functions $t_{i,j} : U_i\cap U_j \rightarrow G$, where $\{U_i\}$ is an open cover of $B$ locally trivializing the bundle.
Analogously, for a simplicial set $B$ and a simplicial group $G$, a $G$-principle bundle over $B$ can be given by a twisting function $r$ which is a collection of maps $r_q : B_q \rightarrow G_{q-1}$.
The two concepts are analogous. The analogy makes me think of basic ideas of Čech methods. But, I could not find details about it written anywhere.
