I know Kan condition (see http://en.wikipedia.org/wiki/Kan_fibration) is something like homotopy extension condition, and I know this condition ensures homotopy defined by the naive idea to be an equivalence condition. But this condition rules out some easy-constructed simplicial sets, like those obtained from the standard simplex, and simplicial sets satisfying this condition are always quite huge.

My question is: can we avoid Kan condition by considering ALL simplicial sets and define homotopy to be an equivalence condition by ARTIFICIALLY adjoining all transitivity? Can we define well-behaved homotopy groups then for all simplicial sets and still get the equivalence between its homotopy category and that of CW complex?