It is known (cf. Lurie's book [*Higher Topos Theory*](http://www.math.harvard.edu/~lurie/papers/highertopoi.pdf) for instance) that higher ($\infty$-) category, in particular topological higher ($\infty$-) groupoids are "better" defined as *weak Kan complexes*, aka *quasi-categories*. Let me recall that a simplicial space $X\_\bullet$ is *weak Kan* if every map $\Lambda\_i^n\longrightarrow X$ (where $\Lambda\_i^n$ is the *horn*) can be extended to a map $\Delta^n\longrightarrow X$ for all $0< i < n$. 


My problem is the following. Let us consider the *fundamental* $(\infty,0)$-groupoid $\pi\_{\leq \infty}X$ of a nice space $X$: $X$ is the set of objects, morphisms between objects are maps $f:[0,1]\longrightarrow X$, $2$-morphisms are homotopies between morphisms, and so on. I am quite confused about how does one, "geometrically", see $\pi\_{\leq \infty}X$ as a simplicial space; for an element of $\pi\_2X$, for instance, has four edges, and hence four different face maps. 



Infinity-categories vs Kan complexes
I suspect your confusion arises in part because homotopies of paths are continuous maps $I^2 \to X$, while 2-morphisms in $\pi\_{\lt \infty} X$ are continuous maps $\Delta^2 \to X$. That is, 2-morphisms are not strictly the same as homotopies of paths.


The dictionary between the two structures is not too bad:


1. A 2-morphism in $\pi\_{\lt \infty} X$ is a homotopy of paths, where either the beginning or the end is fixed (or perhaps it is a homotopy to or from a constant path).
2. A continuous map $I^2 \to X$ can be viewed as a composite of two 2-morphisms in $\pi\_{\lt \infty} X$. You end up using the diagonal $(0,0) \to (1,1)$ in the square to separate the two 2-morphisms, because the simplicial structure of $\Delta^1 \times \Delta^1$ has a diagonal 1-simplex.


I personally find it rather magical that among the sixteen 2-simplices in $\Delta^1 \times \Delta^1$, a pair of them pops out as non-degenerate - it is a rewarding computation.


More generally, higher dimensional cubes, viewed as products of the simplicial set $\Delta^1$, have canonical decompositions into nondegenerate simplices. The corresponding higher homotopies are composites of homotopies with various pieces held constant.


I think this should account for the discrepancy you see in the number of face maps.


