# 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 ${S}_{•}\left(X\right)$ 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, ${S}_{•}\left(X\right)$, having

${S}_{n}\left(X\right)={\mathrm{Hom}}_{\mathrm{Top}}\left({\Delta }_{\mathrm{Top}}^{n},X\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $\Pi \left(X\right)$ 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 adjointgeometric 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.

Revised on September 3, 2012 18:11:18 by Urs Schreiber (131.174.188.82)