TITLE: Kan condition in simplicial homotopy theory
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: You asked: 'can we avoid Kan condition by considering ALL simplicial sets and define homotopy to be an equivalence condition by ARTIFICIALLY adjoining all transitivity?' Why not do things as follows?
To turn homotopy into an equivalence relation you have to not only generate under transitivity but also to add inverses, so to do that you have to  form the free groupoid ... doh! This has been done systematically by Dwyer and Kan (and also by Joyal and Tierney at about the same time) by forming the loop groupoid on the simplicial set, cf. https://ncatlab.org/nlab/show/Dwyer-Kan+loop+groupoid. Of course this ends up with a simplicially enriched groupoid (aka simplicial groupoid) and yes, of course, you can use this to calculate homotopy groups, etc. via the Moore complex construction. ... and yes of course, you can go back to simplicial sets via the loop-groupoid functors left adjoint $\overline{W}$... and low and  behold that yields a Kan complex (and the fillers have the sort of properties you would like in answer to your question ... (although this is rarely shown in texts). This has an advantage in as much as it comes with a lot of fibre bundle / fibration intuitions analogous to those for the loop group in topology.  These are classical, very neat and very useful!