nLab
singular simplicial complex

Idea

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 $S_{•}(X)$ of a topological space $X$ is the nerve of $X$ with respect to the standard cosimplicial topological space ${\Delta }_{\mathrm{Top}}:\Delta \to \mathrm{Top}$. It is thus the simplicial set, $S_{•}(X)$, having

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

Properties

Together with its adjoint – geometric realization $\mid -\mid :\mathrm{sSet}\to \mathrm{Top}$ – the functor $\mathrm{Sing}:\mathrm{Top}\to \mathrm{sSet}$ 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.

Related concepts

• singular simplex

• fundamental infinity-groupoid

• singular homology