Quillens higher K-groups of rings can be realized as πnK(C) - the Waldhausen K-Theory of a suitable Waldhausen category C. Is this also true for Milnor K-Theory of Rings? Is there a functor F from rings to waldhausen categories s.t. $K^M\_n(R)\cong \pi\_n(K(F(R))$?



Does Milnor K-Theory arise from Waldhausen K-Theory
I don't know if there any evidence for this to be true. Note that Quillen K-groups *are defined* as homotopy groups of some space (+-construction, Q-construction, Waldhausen construction etc), whereas Milnor K-groups were defined in terms of generators and relations,
which generalize generators and relations for classical K\_2.


More invariantly Milnor K-groups can be constructed using homology of GL\_n (paper of Suslin and Nesterenko) or as certain motivic cohomology groups of a field (Suslin-Voevodsky).
However, these constructions are unrelated to any homotopy groups.


Also, I'm not sure how you define Milnor K-theory for a general ring R?
(I was interpreting your question with "ring R" replaced by "field F".)


