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# Contents

## 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.

### Relation to geometric realization

Together with its adjointgeometric 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 ordinary (co)homology

Forming from the singular simplicial complex $Sing(X)$ first the free simplicial abelian group $\mathbb{Z}[Sing(X)]$ 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 $X$.

Last revised on June 6, 2020 at 10:21:38. See the history of this page for a list of all contributions to it.