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.