Kan fibrant replacement for a sphere To compute the simplicial homotopy group of a space $X$, we find a Kan fibrant replacement $X \to Y$ and calculate for that for $Y$, which can be implemented in a computer program.
Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibrant replacement.
Could you please show why it's hard, for the easiest possible nontrivial case?
 A: Any fibrant replacement for $S^n$, $n \geq 1$  is going to have infinitely many non-degenerate simplices. This is simply because there are infinitely many elements of $\pi_nS^n$.  So, even though to a mathematician it seems that we can "compute" a fibrant replacement, it is not actually easy to program it in such a way that we can determine the homotopy groups.
A: 
Computing homotopy groups for spheres are fundamentally hard, and I believe the problem lies in the difficulty of finding their Kan fibrant replacement.

Computing the fibrant replacement for simplicial sets is quite
easy: it is given by the Kan fibrant replacement functor Ex^∞.  Explicitly, n-simplices in the fibrant replacement of a simplicial set X are maps Sd^k Δ^n → X, for some k≥0.
Here Sd^k denotes the k-fold barycentric subdivision of a simplicial set.
We allow to increase k by further subdividing, this does not change the simplex.
This description is very simple and can be easily programmed into a computer.
The problem is, however, is that the number of simplices grows exponentially
with k, and we also do not have an efficient way to get an a priori upper
bound for k.  So some problems are bound to be computationally undecidable,
such as the problem of computing whether π_1 of a simplicial set is trivial or not.
