I'm trying to understand the concept of Kan simplicial sets and simplicial groups in particular. My question is, in a nutshell:
What is the relation between the group operation in a simplicial group and the (possible) underlying topological group structure?
For example, Eilenberg-MacLane spaces have a natural simplicial group model, but I don't know whether this correspond to a group operation on the underlying topological space. Does it?
Conversely, can Lie groups be converted into simplicial groups (in a more feasible way than applying the singular ss functor)? I was thinking about $S^3$, the group of unit quaternions. As far as I know, a simplicial set describing $S^3$ up to homotopy type is its Postnikov tower, build as a twisted product of Eilenberg-Maclane spaces, but what about the group operation? Can it be defined so that it reflect somehow the group operation in $S^3$ (at least "up to homotopy" in some way that I cannot formalize yet)?