nLab
singular simplicial complex

The singular simplicial complex $S_{•}(X)$ of a topological space $X$ is the nerve of $X$ with respect to the standard cosimplicial topological space ${\Delta }_{\mathrm{Top}}:\Delta \to \mathrm{Top}$:

$$S_n(X)={\mathrm{Hom}}_{\mathrm{Top}}({\Delta }_{\mathrm{Top}}^n,X).$$S_n(X) = Hom_{Top}(\Delta_{Top}^n, X) \,.

This is always a Kan complex and as such has the interpretation of the fundamental ∞-groupoid $\Pi (X)$ of $X$.