It is known (cf. Lurie's book Higher Topos Theory 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.
Best Answer
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:
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).
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.