I'm reading Segal's paper Classifying Spaces and Spectral Sequences and in section $4$ he tries to prove for good coverings(which admits a partition of unity) $\{U_{\alpha }\} $ over a topological space $X$ the natural projection $BR_U\to X$ is a homotopy equivalence.
The setting goes as follows: given any open cover $\{U_i \} $ of $X$, we define $R_U$ to be the category with objects finite intersections of $U_i $s, and morphisms inclusions, and $X_U $ to be the topological category of pairs $(U_i ,x)$ with $x\in U_i $. Then we have a natural forgetful functor $X_U\to R_U $ which induces a map for classifying spaces $BX_U\to BR_U $ which is surjective and fibres are simplices.
In Segal's proof he only shows the homotopy equivalence between $X$ and $BX_U $, so I wonder if $BX_U\to BR_U $ is a homotopy equivalence. I guess since fibres are contractible we have the weak homotopy equivalence $BX_U\to BR_U $, and if we could show it is a fibration then by Whitehead's theorem we will have this to be a homotopy equivalence, but I don't know if it is actually the case.