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 $S_\bullet(X)$ 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.)

Properties

Preservation of model structure

The singular complex functor preserves all five classes of maps in a model category: weak equivalences, cofibrations, acyclic cofibrations, fibrations, and acyclic fibrations.

• Weak equivalences are preserved; the reason depends on how the model structures are constructed. The classical model structure on topological spaces can be defined to be transferred from simplicial sets, in which case the singular complex preserves weak equivalences by definition. More traditionally, the weak equivalences in the classical model structure on simplicial sets are defined to be those whose geometric realization is a (weak) homotopy equivalence, and the preservation then follows from the 2-out-of-3 property and the fact that the geometric realization of the total singular complex is a CW replacement? that is weakly homotopy equivalent to the original space.

• Cofibrations are preserved because all cofibrations of topological spaces are injective functions, injective maps are preserved, and all injective maps of simplicial sets are cofibrations.

• Fibrations and acyclic fibrations are preserved because the singular complex functor is a right Quillen functor. More explicitly, this is because geometric realization takes the generating cofibrations and acyclic cofibrations of simplicial sets to those of topological spaces.

Relation to geometric realization

Together with its adjoint—geometric realization $|-| : sSet \to Top$—the functor $Sing : Top \to sSet$ 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.

Relation to singular homology

Forming from the singular simplicial complex $S_\bullet(X)$ first the free simplicial abelian group $\mathbb{Z}[S_\bullet(X)]$ and then under the Dold-Kan correspondence the corresponding normalized chain complex yields the chain complex of normalized singular chains, which computes the singular homology of $X$.

Related concepts

• singular simplex

• fundamental infinity-groupoid

• singular homology