This construction ‘probes’ a space $X$ by mapping geometric simplices into it. It is one of the classical approaches to determining invariants of the homotopy type of the space.

Definition

The singular simplicial complex$S_\bullet(X)$ of a topological space$X$ is the nerve of $X$ with respect to the standard cosimplicial topological space $\Delta_{Top} : \Delta \to Top$. It is thus the simplicial set, $S_\bullet(X)$, having

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

as its set of $n$-simplices, and fairly obvious faces and degeneracy mappings obtains by restriction along the structural maps of $\Delta_{Top} : \Delta \to Top$. This is always a Kan complex and as such has the interpretation of the fundamental ∞-groupoid$\Pi(X)$ of $X$.

The $n$-simplices of this are just singular n-simplices generalising paths in $X$. (The term -singular_ is used because there is no restriction that the continuous function used should be an embedding, as would be the case in, for instance, a triangulation where a simplex in the underlying simplicial complex corresponds to an embedding of a simplex.)