The category of simplices $\Delta$ has a terminal object $[0]$, hence its nerve is contractible. What can be said about the nerve of its subcategory $\Delta\_{\mathrm{mono}}$ which contains only the coface maps?



Nerve of the semi-simplex category
Let $C: \Delta\_{\mathrm{mono}}\to \Delta\_{\mathrm{mono}}$ be the "cone" functor, given on objects by $C([p])=[p+1]$, and on morphisms by $C(\delta)(0)=0$ and $C(\delta)(i) = \delta(i-1)+1$. Then there are natural monomorphisms
$$[p] \to C([p]) \leftarrow [0]$$
which give a zig-zag of natural transformations relating the identity functor on $\Delta\_{\mathrm{mono}}$ to a constant functor. So its got contractible nerve.


