This construction ‘probes’ a space by mapping geometric simplices into it. It is one of the classical approaches to determining invariants of the homotopy type of the space.
as its set of -simplices, and fairly obvious faces and degeneracy mappings obtains by restriction along the structural maps of . This is always a Kan complex and as such has the interpretation of the fundamental ∞-groupoid of .
The -simplices of this are just singular n-simplices generalising paths in . (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.)
Together with its adjoint – geometric realization – the functor is part of the Quillen equivalence between the model structure on topological spaces and the model structure on simplicial sets that is sometimes called the homotopy hypothesis-theorem.
Forming from the singular simplicial complex first the free simplicial abelian group and then under the Dold-Kan correspondence the correspondin normalized chain complex yields this chain complex which computes the singular homology of .