TITLE: Why study simplicial homotopy groups?
QUESTION [19 upvotes]: The standard definition for simplicial homotopy groups only works for Kan complexes (cf. http://ncatlab.org/nlab/show/simplicial+homotopy+group). I learned that the hard way, when I tried to compute a very simple example, i.e. the homotopy group of the boundary of the standard 2-simplex. 
My naive idea to actually compute simplicial homotopy groups for arbitrary simplicial sets was taking the fibrant replacement. But obviously we need a model structure for that. Then again, a weak equivalence in the usual model structure for simplicial sets is precisely a weak equivalence of the geometric realization.(cf. http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets) 
As I understand it so far, the only satisfactory way to talk about simplicial homotopy groups requires the notion of "classical" homotopy groups. Hence my question: why does it still make sense, to actually talk about simplicial homotopy groups in the first place?

REPLY [6 votes]: Well, both in the book of Peter May and the paper from Curtis it is proved that the homotopy category of the category of Kan simplicial sets is equivalent to the homotopy category of CW complexes. The equivalence is given by the adjoint functors "geometric realisation" and "total singular simplicial set". This is one of many motivations to study simplicial sets as a very suitable "discrete model" for topological spaces. One very nice thing about simplicial sets is how constructive things are (often :) ). As far as I have read, simplicial sets have been first used for constructing classifying spaces for topological groups.
The fact that we can define the homotopy groups within the category of Kan simplicial sets is great, because this gives us the opportunity to study them using some combinatorial tools as well. You can check this paper
http://arxiv.org/abs/1211.3093 
Here the authors develop a polynomial time algorithm, which computes homotopy groups of simply connected simplicial complexes. For example, starting with a finite simplicial complex model of the 2-sphere, we can introduce an ordering of the vertices and generate a simplicial set. This simplicial set will never be Kan, but using Postnikov towers and a suitable model for Eilenberg-MacLane spaces, one can build up Kan simplicial sets, which approximate it homotopically and use the Postnikov stages for the computations. The result is beautiful and very constructive.
I hope this gives some motivation for the study of simplicial homotopy groups.