Why study simplicial homotopy groups?

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?

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The can also be computed as the homology of a non-abelian chain complex: the Moore complex on Kan's loop simplicial group (which is defined from the simplicial set). This works in the connected case. Otherwise groupoids are necessary. –  Fernando Muro Nov 10 '11 at 15:13
Kan computes $\pi_3(S^2)$ via the Moore complex. It takes many pages. The idea is that a subgroup of free group (which is of course free), has a system of generators that is described by the Nielsen-Schrier theorem. –  John Klein Nov 11 '11 at 0:52

To compute the homotopy groups of a simplicial set, you need to be able to construct a weak equivalence $X \to Y$ where $Y$ is a Kan complex, and then compute the homotopy groups of $Y$ using the definitions you were discussing.

This might seem circular - you need to detect if $X \to Y$ is an equivalence. However, you can construct $Y$ directly using certain more elementary equivalences. Specifically, for a map of a horn $\Lambda \to X$ we can form the pushout of the diagram $\Delta \leftarrow \Lambda \rightarrow X$, called $X'$; on geometric realizations this is homotopy equivalent because you can construct an explicit retraction. The class of maps $X \to Y$ generated by such pushouts is called the family of anodyne extensions of $X$, and you can always construct an anodyne extension which is a Kan complex by the small object argument (you keep gluing in solutions to every possible horn-filling problem).

If you want a more canonical answer, there is also Kan's $Ex^\infty$ construction.

If you want another reference, there are Kan's older papers, and my recollection is that Joyal and Tierney has quite a number of details as well.

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If $X$ is a simplicial set which is not Kan, you can compute the homotopy groups of $X$ by choosing a weak homotopy equivalence $f: X \rightarrow Y$ where $Y$ is Kan and then applying the construction you are familiar with to $Y$. There are many ways of characterizing the relationship between $X$ and $Y$ without every mentioning topology or model categories (though I'm not sure it is so helpful to avoid these). For example, you can take any map $f: X \rightarrow Y$ with the following property: for every Kan complex $Z$, composition with $f$ induces a bijection from $[Y,Z]$ to $[X,Z]$, where $[K,Z]$ denotes the set of maps from $K$ into $Z$ up to (simplicial) homotopy. There are several purely combinatorial constructions of $Y$ from $X$: for example, Kan's $Ex^{\infty}$ functor.

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