nLab singular cohomology

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.

…

Related concepts

• cochain on a simplicial set

• cup product

• de Rham theorem

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

• Kan complex

• ∞-groupoid.

References

Early references on (co)homology

The original references on chain homology/cochain cohomology and ordinary cohomology in the form of cellular cohomology:

• Andrei Kolmogoroff, Über die Dualität im Aufbau der kombinatorischen Topologie, Recueil Mathématique 1(43) (1936), 97–102. (mathnet)

A footnote on the first page reads as follows, giving attribution to Alexander 35a, 35b:

Die Resultate dieser Arbeit wurden für den Fall gewöhnlicher Komplexe vom Verfasser im Frühling und im Sommer 1934 erhalten und teilweise an der Internationalen Konferenz für Tensoranalysis (Moskau) im Mai 1934 vorgetragen. Die hier dargestellte allgemeinere Theorie bildete den Gegenstand eines Vortrages, den der Verfasser an der Internationalen Topologischen Konferenz (Moskau, September 1935) hielt; bei letzterer Gelegenheit erfuhr er, dass ein grosser Teil dieser Resultate im Falle von Komplexen indessen von Herrn Alexander erhalten worden ist. Vgl. die inzwischen erschienenen Noten von Herrn Alexander in den «Proceedings of the National Academy of Sciences U.S.A.», 21, (1935), 509—512. Herr Alexander trug über seine Resultate ebenfalls an der Moskauer Topologischen Konferenz vor. Verallgemeinerungen für abgeschlossene Mengen und die Konstruktion eines Homologieringes für Komplexe und abgeschlossene Mengen, über welche der Verfasser ebenso an der Tensorkonferenz 1934 vorgetragen hat, werden in einer weiteren Publikation dargestellt. Diese weitere Begriffsbildungen sind übrigens ebenfalls von Herrn Alexander gefunden und teilweise in den erwähnten Noten publiziert.

• Andrei Kolmogoroff, Homologiering des Komplexes und des lokal-bicompakten Raumes, Recueil Mathématique 1(43) (1936), 701–705. mathnet.

• J. W. Alexander, On the chains of a complex and their duals, Proc. Nat. Acad. Sei. USA, 21(1935), 509–511 (doi:10.1073/pnas.21.8.50)

• J. W. Alexander, On the ring of a compact metric space, Proc. Nat. Acad. Sci. USA, 21 (1935), 511–512 (doi:10.1073/pnas.21.8.511)

• J. W. Alexander, On the connectivity ring of an abstract space, Ann. of Math., 37 (1936), 698–708 (doi:10.2307/1968484, pdf)

The term “cohomology” was introduced by Hassler Whitney in

• Hassler Whitney, On matrices of integers and combinatorial topology. Duke Mathematical Journal 3:1 (1937), 35–45 (doi:10.1215/S0012-7094-37-00304-1)

See also

• Hassler Whitney, On products in a complex, Annals of Math. 39 (1938) 397–432 (doi:10.2307/1968795)

The notion of singular cohomology is due to

• Samuel Eilenberg, Singular homology theory, Annals of Mathematics 45:3 (1944) (doi:10.2307/1969185)

The notion of monadic cohomology via canonical resolutions:

• Michael Barr, Jon Beck, Homology and Standard Constructions, in: Seminar on Triples and Categorical Homology Theory, Lecture Notes in Maths. 80, Springer (1969), Reprints in Theory and Applications of Categories 18 (2008) 186-248 [tac:18, pdf]

• Michael Barr, Cartan-Eilenberg cohomology and triples, J. Pure Applied Algebra 112 3 (1996) 219–238 [doi:10.1016/0022-4049(95)00138-7, pdf, pdf]

• Michael Barr, Algebraic cohomology: the early days, in Galois Theory, Hopf Algebras, and Semiabelian Categories, Fields Institute Communications 43 (2004) 1–26 [doi:10.1090/fic/043, pdf, pdf]

The general abstract perspective on cohomology (subsuming sheaf cohomology, hypercohomology, non-abelian cohomology and indications of Whitehead-generalized cohomology) was essentially established in:

• Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology (1973)

General

Relation to sheaf cohomology:

• Yehonatan Sella, Comparison of sheaf cohomology and singular cohomology (arXiv:1602.06674)

A simplified proof using hypercovers can be found in

• Dan Petersen, A remark on singular cohomology and sheaf cohomology (arXiv:2102.06927)