Contents

# Contents

## Definition

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

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

$H^n(X,\mathbb{Z}) := \pi_0 \infty Grpd(\Pi(X), \mathcal{B}^n \mathbb{Z}) \,.$

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

$\Pi(X) = Sing X = Hom_{Top}(\Delta^\bullet_{Top}, X) \,,$

the singular simplicial complex of $X$.

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

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

which is

• the underlying simplicial set under the forgetful functor

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

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

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

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

concentrated in degree $n$.

So in this model we have

$H^n(X,\mathbb{Z}) = \pi_0 sSet(Sing X, U(\Xi \mathbb{Z}[n])) \,.$

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

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

$\cdots \simeq \pi_0 sAb( Ch_n(X), \Xi \mathbb{Z}[n] ) \,,$

where

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

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

$\cdots \simeq \pi_0 sAb( \mathbb{Z}[Sing X], \Xi \mathbb{Z}[n] ) \,.$

Using next the Dold-Kan adjunction this is

$\cdots \simeq H_0 Ch( Ch_\bullet(X), \mathbb{Z}[n] ) \,,$

where

$Ch_\bullet(X) := N^\bullet(\mathbb{Z}(Sing X))$

is the Moore complex of normalized chains of $\mathbb{Z}[Sing X]$: 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.

## Comparison to sheaf cohomology

If the topological space $X$ is semi-locally contractible (meaning: any open subset $U\subset X$ has an open cover $W$ by open subsets $W_i\subset U$ that are contractible in $U$), then the sheaf cohomology of $X$ is isomorphic to the singular cohomology of $X$ for any abelian group of coefficients.

This was proved in (Sella 16).

## 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

## References

Last revised on October 26, 2017 at 13:32:33. See the history of this page for a list of all contributions to it.