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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

2 Answers 2

up vote 4 down vote accepted

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.

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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.

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