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

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.

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What is the question? –  S. Carnahan Sep 6 '10 at 14:52

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 $C[Mor(C)^{-1}]$ at all arrows, then by taking $Hom(a,a) = Aut(a)$ in this groupoid. There are size issues, clearly, but for essentially small $C$ these can be ignored.