This MO answer and its comments, suggest a cool characterization of paracompactness I have never seen before.
For a space $B$ let $(U_i)$ be an open cover. Form the groupoid $\amalg_iU_i\times_B \amalg_iU_i \rightrightarrows \amalg_iU_i$ and look at its classifying space $\mathbf B U$. If I understand correctly, this comment says the following are equivalent:
- B is paracompact;
- $B$ is a deformation retract of $\mathbf BU$ for any open cover.
Unfortunately, I can't fill in the proof in any direction. Are the above conditions equivalent? How to prove this?