The classifying space of a category $\scr{C}$ is obtained by taking its nerve $N\scr{C}$, which is the simplicial set defined by $$ N\mathscr{C}_n:= \mathrm{Fun}([n],\mathscr{C}) $$ and the classifying space is defined as $$ B\mathscr{C}:= |N\mathscr{C}| $$ the geometric realization of the nerve. The only concrete examples I have every played with are the classifying spaces of groups, $BG$. But these all end up being $K(G,1)$'s.
Question: What is an explicit example of a category $\mathscr{C}$ so that its classifying space has nontrivial higher homotopy groups.
I know that such things should exist; it is my understanding that Quillen's Q-construction takes a category $M$ and outputs a category $QM$ whose classifying space is the K-theory $K(M)$.
Thanks!
For a very simple example, consider the poset $\{a,b,c,d,e,f\}$ with $a,b\leq c,d\leq e,f$. The classifying space of this poset is homeomorphic to $S^2$ (you can explicitly list out all the nondegenerate simplices in its nerve and draw a picture of them), which has plenty of higher homotopy groups. More generally, in fact, every simplicial complex is homeomorphic to the classifying space of a poset, namely its poset of faces (indeed, the nerve of the poset of faces is just the barycentric subdivision of the simplicial complex you started with). Since every space is weak homotopy equivalent to a simplicial complex, this means that every (weak) homotopy type can be the realized as the classifying space of a category.