# nLab singular cohomology

### Context

#### Topology

topology

algebraic topology

cohomology

# Contents

## Definition

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

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

${H}^{n}\left(X,ℤ\right):={\pi }_{0}\infty \mathrm{Grpd}\left(\Pi \left(X\right),{ℬ}^{n}ℤ\right)\phantom{\rule{thinmathspace}{0ex}}.$H^n(X,\mathbb{Z}) := \pi_0 \infty Grpd(\Pi(X), \mathcal{B}^n \mathbb{Z}) \,.

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

$\Pi \left(X\right)=\mathrm{Sing}X={\mathrm{Hom}}_{\mathrm{Top}}\left({\Delta }_{\mathrm{Top}}^{•},X\right)\phantom{\rule{thinmathspace}{0ex}},$\Pi(X) = Sing X = Hom_{Top}(\Delta^\bullet_{Top}, X) \,,

the singular simplicial complex of $X$.

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

${ℬ}^{n}ℤ=U\left(\Xi ℤ\left[n\right]\right)$\mathcal{B}^n \mathbb{Z} = U (\Xi \mathbb{Z}[n])

which is

• the underlying simplicial set under the forgetful functor

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

from abelian simplicial groups to simplicial sets;

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

$\mathrm{sAb}\stackrel{\stackrel{\Xi }{←}}{\underset{}{\to }}{\mathrm{Ch}}^{+}$sAb \stackrel{\overset{\Xi}{\leftarrow}}{\underset{}{\to}} Ch^+
• of the chain complex

$ℤ\left[n\right]=\left(\cdots \to ℤ\to 0\to 0\to \cdots \to 0\right)$\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

${H}^{n}\left(X,ℤ\right)={\pi }_{0}\mathrm{sSet}\left(\mathrm{Sing}X,U\left(\Xi ℤ\left[n\right]\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$H^n(X,\mathbb{Z}) = \pi_0 sSet(Sing X, U(\Xi \mathbb{Z}[n])) \,.

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

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

$\cdots \simeq {\pi }_{0}\mathrm{sAb}\left({\mathrm{Ch}}_{n}\left(X\right),\Xi ℤ\left[n\right]\right)\phantom{\rule{thinmathspace}{0ex}},$\cdots \simeq \pi_0 sAb( Ch_n(X), \Xi \mathbb{Z}[n] ) \,,

where

$F\left(\mathrm{Sing}X\right)=ℤ\left[\mathrm{Sing}X\right]$F(Sing X) = \mathbb{Z}[Sing X]

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

$\cdots \simeq {\pi }_{0}\mathrm{sAb}\left(ℤ\left[\mathrm{Sing}X\right],\Xi ℤ\left[n\right]\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq \pi_0 sAb( \mathbb{Z}[Sing X], \Xi \mathbb{Z}[n] ) \,.

Using next the Dold-Kan adjunction this is

$\cdots \simeq {H}_{0}\mathrm{Ch}\left({\mathrm{Ch}}_{•}\left(X\right),ℤ\left[n\right]\right)\phantom{\rule{thinmathspace}{0ex}},$\cdots \simeq H_0 Ch( Ch_\bullet(X), \mathbb{Z}[n] ) \,,

where

${\mathrm{Ch}}_{•}\left(X\right):={N}^{•}\left(ℤ\left(\mathrm{Sing}X\right)\right)$Ch_\bullet(X) := N^\bullet(\mathbb{Z}(Sing X))

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

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

## 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 $\infty$-groupoids at

Revised on October 29, 2013 09:16:50 by Adeel Khan (132.252.63.205)