Suppose I have a topological category $\mathcal{C}$ of the following form: The object space consists of just two points $p\_1, p\_2$. The endomorphism space of $p\_1$ contains just the identity. The endomorphism space $End(p\_2)$ of $p\_2$ is a group $G$ and we have a restriction map $End(p\_2) \to hom(p\_1, p\_2)$ which is a homotopy equivalence. 



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My guess would be that it is either homotopy equivalent to $BG$ or contractible or something weird like $BG \times G$. Note that even though $hom(p\_1,p\_2)$ is a $G$-space, $B\mathcal{C}$ does not seem to be the bar construction $B(G,G,\ast)$ or is it? For the latter, the space of objects should be the $G$-space in question and not just two points. Moreover, note that $p\_1$ is not necessarily an initial object, since the space $hom(p\_1,p\_2)$ is generally non-trivial. There also is an obvious map $BG \to B\mathcal{C}$, but I am unable to show that it is a homotopy equivalence, so maybe it isn't.



classifying space of a strange category
I think it is equivalent to $BG$. Picking an $x\_0 \in X := hom(p\_1, p\_2)$ we get a map


$$X \to G$$


sending x\_0 \* g to g. Using this on morphism sets we get a map $F: \mathcal{C} \to G$ which you easily check to be a functor. Furthermore, the element $x\_0$ can be used to get a natural transformation $id\_\mathcal{C} \Rightarrow inc \circ F$.


