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In Hatcher's book on algebraic topology (p 89) the universal bundle $EG$ ($G$ discrete group) carries structure of a $\triangle$-complex whose $n$-simplices are the ordered $(n + 1)$ tuples $[g_0, ... ,g_n]$ of elements of $G$. As quotient space the classifing space $BG=EG/G$ inherits structure $\triangle$-complex where a $n$-simplex $BG$ can be written uniquely in the form $[g_1 \vert g_2\vert ... \vert g_n]:= G \cdot [e_G, g_1, g_1g_2,..., g_1,.., g_n]$.

My question is if there is a natural way to choose a trivializing cover $\{U_i\}_{i \in I}$ of the universal principal $G$-bundle $EG \to BG$, ie a family of open subsets of $BG$ such that $EG \vert \_{U_i} \cong U_i \times G$ and what are the the associated transition functions $h_{ij}: U_i \cap U_j \to G$ used for patching functions $ U_i \cap U_j \times G \to U_i \cap U_j \times G, (u,g) \mapsto (u, h_{ij}(u)g)$.

The question is closely connected to a problem appearing in a construction here

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