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. 



Definition for fundamental group (higher homotopy groups) for a category?
In this definition BQM can be taken to be a space - the [geometric realization](http://en.wikipedia.org/wiki/Simplicial_set#Geometric_realization) of the [nerve](http://en.wikipedia.org/wiki/Nerve_%2528category_theory%2529) 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.


