Contents

Definition

The singular cohomology of a topological space $X$ is the cohomology in ∞Grpd of its fundamental ∞-groupoid $\Pi (X)$:

for ${ℬ}^n\mathbb{Z}\in \mathrm{\infty }\mathrm{Grpd}$ the Eilenberg-MacLane object with the group $\mathbb{Z}$ in degree $n$, the degree $n$-singular cohomology of $X$ is

With $\mathrm{\infty }\mathrm{Grpd}$ presented by the category sSet of simplicial sets, the fundamental $\mathrm{\infty }$-groupoid $\Pi (X)$ is modeled by the Kan complex

the singular simplicial complex of $X$.

The object ${ℬ}^n\mathbb{Z}$ is usefully modeled by the simplicial set

which is

• the underlying simplicial set under the forgetful functor

  $$(F⊣U)\mathrm{sAb}\stackrel{\stackrel{F}{\leftarrow }}{\underset{U}{\to }}\mathrm{sSet}$$(F \dashv U) sAb \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet

  from abelian simplicial groups to simplicial sets;

• of the abelian simplicial group $\Xi \mathbb{Z}[n]$ which is the image under the Dold-Kan correspondence

  $$\mathrm{sAb}\stackrel{\stackrel{\Xi }{\leftarrow }}{\underset{}{\to }}{\mathrm{Ch}}^{+}$$sAb \stackrel{\overset{\Xi}{\leftarrow}}{\underset{}{\to}} Ch^+

• of the chain complex

  $$\mathbb{Z}[n]=(\cdots \to \mathbb{Z}\to 0\to 0\to \cdots \to 0)$$\mathbb{Z}[n] = (\cdots \to \mathbb{Z} \to 0 \to 0 \to \cdots \to 0)

  concentrated in degree $n$.

So in this model we have

A cocycle in this cohomology theory is a cochain on a simplicial set, on the singular complex $\mathrm{Sing}X$.

Using the adjunction $(F⊣U)$ this is isomorphic to

where

is the free abelian simplicial group on the simplicial set $\mathrm{Sing}X$: this is the simplicial abelian group of singular chains of $X$. Its elements are formal sums of continuous maps ${\Delta }_{\mathrm{Top}}^n\to X$. In this form

Using next the Dold-Kan adjunction this is

where

is the Moore complex of normalized chains of $\mathbb{Z}[\mathrm{Sing}X]$: this is the complex of singular chains, formal sums over $\mathbb{Z}$ of simplices in $X$.

This way singular cohomology is the abelian dual of singular homology.

…

Related concepts

• cochain on a simplicial set

• cup product

• de Rham theorem

Discussion

A previous version of this entry led to the following discussion, which later led to extensive discussion by email. Partly as a result of this and similar discussions, there is now more information on how Kan complexes are $\mathrm{\infty }$-groupoids at

• Kan complex

• ∞-groupoid.