Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I expect it to be contractible.
However, I was not able to explicitly proof contractibility starting from the definition $$B(X,\le)=(\coprod_{i\in\mathbb{N}_0}N_i(X,\le)\times\Delta^i)/\tilde{} $$
Can anyone help me?