Definition for fundamental group (higher homotopy groups) for a category?

Question

How to define homotopy groups in categories as in Quillen's definition for Higher algebraic K-theory: K_i(M)=\pi_{i+1}(BQM, 0), where M is a small category and BQM is the classifying space of QM. thank you.

Answer 1

In this definition BQM can be taken to be a space - the geometric realization of the nerve of the category QM. The homotopy groups are then the usual homotopy groups from topology.

There also is a definition of the homotopy group of a simplicial set - you can thus compute the homotopy group of the nerve without passing to the geometric realization first - and the definition you give looks more like that, but the answers are isomorphic.

Answer 2

Quillen shows at the beginning of his article on higher algebraic K-theory that you can calculate the fundamental group $\pi_1(C,a)$ of a category $C$ at an object $a$ by forming the localisation $C[\operatorname{Mor}(C)^{-1}]$ at all arrows, then by taking $\operatorname{Hom}(a,a) = \operatorname{Aut}(a)$ in this groupoid. There are size issues, clearly, but for essentially small $C$ these can be ignored.