nLab chain homology and cohomology

Idea

In the context of homological algebra, for $V_\bullet \in Ch_\bullet(\mathcal{A})$ a chain complex, its chain homology group in degree $n$ is akin to the $n$-th homotopy groups of a topological space. It is defined to be the quotient of the $n$-cycles by the $n$-boundaries in $V_\bullet$.

Dually, for $V^\bullet \in Ch^\bullet(\mathcal{A})$ a cochain complex, its cochain cohomology group in degree $n$ is the quotient of the $n$-cocycles by the $n$-coboundaries.

Basic examples are the singular homology and singular cohomology of a topological space, which are the (co)chain (co)homology of the singular complex.

Chain homology and cochain cohomology constitute the basic invariants of (co)chain complexes. A quasi-isomorphism is a chain map between chain complexes that induces isomorphisms on all chain homology groups, akin to a weak homotopy equivalence. A category of chain complexes equipped with quasi-isomorphisms as weak equivalences is a presentation for the stable (infinity,1)-category of chain complexes.

Definition

Let $\mathcal{A}$ be an abelian category such as that of $R$-modules over a commutative ring $R$. For $R = \mathbb{Z}$ the integers this is the category Ab of abelian groups. For $R = k$ a field, this is the category Vect of vector spaces over $k$.

Write $Ch_\bullet(\mathcal{A})$ for the category of chain complexes in $\mathcal{A}$. Write $Ch^\bullet(\mathcal{A})$ for the category of cochain complexes in $\mathcal{A}$.

We label differentials in a chain complex as follows:

$$V_\bullet = [ \cdots \to V_{n+1} \stackrel{\partial_n}{\to} V_n \to \cdots ]$$

Definition

For $V_\bullet \in Ch_\bullet(\mathcal{A})$ a chain complex and $n \in \mathbb{Z}$, the chain homology $H_n(V)$ of $V$ in degree $n$ is the abelian group

$$H_n(V) \coloneqq \frac{Z_n(V)}{B_n(V)} = \frac{ker(\partial_{n-1})}{im(\partial_n)}$$

given by the quotient (cokernel) of the group of $n$-cycles by that of $n$-boundaries in $V_\bullet$.

Properties

Functoriality

Proposition

For all $n \in \mathbb{N}$ forming chain homology extends to a functor from the category of chain complexes in $\mathcal{A}$ to $\mathcal{A}$ itself

$$H_n(-) : Ch_\bullet(\mathcal{A}) \to \mathcal{A} \,.$$

Proof

One checks that chain homotopy (see there) respects cycles and boundaries.

Proposition

Chain homology commutes with direct product of chain complexes:

$$H_n(\prod_i C^{(i)}) \simeq \prod_i H_n(C^{(i)}) \,.$$

Similarly for direct sum.

Respect for direct sums and filtered colimits

Proposition

The chain homology functor preserves direct sums:

for $A,B \in Ch_\bullet$ and $n \in \mathbb{Z}$, the canonical morphism

$$H_n(A \oplus B) \to H_n(A) \oplus H_n(B)$$

is an isomorphism.

Proposition

The chain homology functor preserves filtered colimits:

for $A \colon I \to Ch_\bullet$ a filtered diagram and $n \in \mathbb{Z}$, the canonical morphism

$$H_n(\underset{\to_i}{\lim} A_i) \to \underset{\to_i}{\lim} H_n(A_i)$$

is an isomorphism.

This is spelled out for instance as (Hopkins-Mathew , prop. 23.1).

Examples

• For $X$ a topological space and $Sing X$ its singular simplicial complex, write $N \mathbb{Z}[Sing X]$ for the normalized chain complex of the simplicial abelian group that is degreewise the free abelian group on $Sing X$. The resulting chain homology is the singular homology of $X$

  $$H_\bullet( N \mathbb{Z}[Sing X]) \simeq H_\bullet(X, \mathbb{Z}) \,.$$

• Koszul homology

• Ext, Tor

In the context of homotopy theory

We discuss here the notion of (co)homology of a chain complex from a more abstract point of view of homotopy theory, using the nPOV on cohomology as discussed there.

A chain complex in non-negative degree is, under the Dold-Kan correspondence a homological algebra model for a particularly nice topological space or ∞-groupoid: namely one with an abelian group structure on it, a simplicial abelian group. Accordingly, an unbounded (arbitrary) chain complex is a model for a spectrum with abelian group structure.

One consequence of this embedding

$$N : Ch_+ \to \infty Grpd$$

induced by the Dold-Kan nerve is that it allows to think of chain complexes as objects in the (∞,1)-topos ∞Grpd or equivalently Top. Every (∞,1)-topos comes with a notion of homotopy and cohomology and so such abstract notions get induced on chain complexes.

Of course there is an independent, age-old definition of homology of chain complexes and, by dualization, of cohomology of cochain complexes.

This entry describes how these standard definition of chain homology and cohomology follow from the general (∞,1)-topos nonsense described at cohomology and homotopy.

The main statement is that

• the naïve homology groups of a chain complex are really its homotopy groups, in the abstract sense (i.e. with the chain complex regarded as a model for a space/$\infty$-groupoid);

• the naïve cohomology groups of a cochain complex are really the abstract cohomology groups of the dual chain complex.

Preliminaries

Before discussing chain homology and cohomology, we fix some terms and notation.

Eilenberg-MacLane objects

In a given (∞,1)-topos there is a notion of homotopy and cohomology for every (co-)coefficient object $A$ ($B$).

The particular case of chain complex homology is only the special case induced from coefficients given by the corresponding Eilenberg-MacLane objects.

Assume for simplicity here and in the following that we are talking about non-negatively graded chain complexes of vector spaces over some field $k$. Then for every $n \in \mathbb{N}$ write

$$\begin{aligned} \mathbf{B}^n k &:= ( \cdots \to \mathbf{B}^n k_n \to \cdots \to \stackrel{\partial}{\to} \mathbf{B}^n k_1 \stackrel{\partial}{\to} \mathbf{B}^n k_0) \\ &= ( \cdots \to k \to \cdots \to 0 \to 0 ) \end{aligned}$$

for the $n$th Eilenberg-MacLane object.

Notice that this is often also denoted $k[n]$ or $k[-n]$ or $K(k,n)$.

Homotopy and cohomology

With the Dold-Kan correspondence understood, embedding chain complexes into ∞-groupoids, for any chain complexes $X_\bullet$, $A_\bullet$ and $B_\bullet$ we obtain

• the $\infty$-groupoid

  $$\mathbf{H}_{\infty Grpd}(X_\bullet, A_\bullet)$$

  whose * objects are the $A$-valued cocycles on $X$; * morphisms are the coboundaries between these cocycles; * 2-morphisms are the coboundaries between coboundaries * etc. and where the elements of $\pi_0 \mathbf{H}(X_\bullet,A_\bullet)$ are the cohomology classes

• the $\infty$-groupoids

  $$\mathbf{H}_{\infty Grpd}(B_\bullet, X_\bullet)$$

  whose * objects are the $B$-co-valued cycles on $X$; * morphisms are the boundaries between these cycles; * 2-morphisms are the boundaries between boundaries * etc. and where the elements of $\pi_0 \mathbf{H}(B_\bullet,X_\bullet)$ are the homotopy classes

Chain homology as homotopy

For $X_\bullet := V_\bullet$ any chain complex and $H_n(V_\bullet)$ its ordinary chain homology in degree $n$, we have

$$H_n(V_\bullet) \simeq \pi_0 \mathbf{H}(\mathbf{B}^n k_\bullet, V_\bullet) \,.$$

A cycle $c : \mathbf{B}^n k_\bullet \to V_\bullet$ is a chain map

$$\array{ \cdots &\stackrel{\partial}{\to}& 0 &\stackrel{\partial}{\to}& k &\stackrel{\partial}{\to}& 0 &\stackrel{\partial}{\to}& \cdots \\ && \downarrow && \downarrow^{c_n} && \downarrow \\ \cdots &\stackrel{\partial}{\to}& V_{n+1} &\stackrel{\partial}{\to}& V_n &\stackrel{\partial}{\to}& V_{n-1} &\stackrel{\partial}{\to}& \cdots }$$

Such chain maps are clearly in bijection with those elements $c_n \in V_n$ that are in the kernel of $V_n \stackrel{\partial}{\to} V_{n-1}$ in that $\partial c_n = 0$.

A boundary $\lambda : c \to C'$ is a chain homotopy

$$\array{ \cdots &\stackrel{\partial}{\to}& 0 &\stackrel{\partial}{\to}& k &\stackrel{\partial}{\to}& 0 &\stackrel{\partial}{\to}& \cdots \\ && & {}^{\lambda_n}\swarrow \\ \cdots &\stackrel{\partial}{\to}& V_{n+1} &\stackrel{\partial}{\to}& V_n &\stackrel{\partial}{\to}& V_{n-1} &\stackrel{\partial}{\to}& \cdots }$$

such that $c' = c + \partial \lambda$.

(…)

Cohomology of cochain complexes

The ordinary notion of cohomology of a cochain complex is the special case of cohomology in the stable (∞,1)- category of chain complexes.

For $V^\bullet$ a cochain complex let

$$\begin{aligned} X &:= V_\bullet = (V^\bullet)^* \\ &= (\cdots \to X_{n+1} \stackrel{\partial}{\to} X_n \stackrel{\partial}{\to} X_{n-1} \to \cdots) \\ & := (\cdots \to V_{n+1} \stackrel{\partial}{\to} V_n \stackrel{\partial}{\to} V_{n-1} \to \cdots) \end{aligned}$$

be the corresponding dual chain complex. Let

$$\begin{aligned} A &:= \mathbf{B}^n I \\ &= (\cdots \to A_{n+1} \to A_n \to A_{n-1} \to \cdots ) \\ & = (\cdots \to 0 \to I \to 0 \to \cdots ) \end{aligned}$$

be the chain complex with the tensor unit (the ground field, say) in degree $n$ and trivial elsewhere. Then

$$\begin{aligned} \mathbf{H}(X,A) &= Ch(V_\bullet, \mathbf{B}^n I) \end{aligned}$$

has

• as objects chain morphisms $c : V_\bullet \to \mathbf{B}^n I$

  $$\array{ \cdots &\to& V_{n+1} &\stackrel{\partial}{\to}& V_{n} &\stackrel{\partial}{\to}& V_{n-1} &\to& \cdots \\ && \downarrow^{c_{n+1}} && \downarrow^{c_{n}} && \downarrow^{c_{n-1}} \\ \cdots &\to& 0 &\stackrel{\partial}{\to}& I &\stackrel{\partial}{\to}& 0 &\to& \cdots } \,.$$

  These are in canonical bijection with the elements in the kernel of $d_{n}$ of the dual cochain complex $V^\bullet = [V_\bullet,I]$.

• as morphism chain homotopies $\lambda : c \to c'$

  $$\array{ \cdots &\to& V_{n+1} &\stackrel{\partial}{\to}& V_{n} &\stackrel{\partial}{\to}& V_{n-1} &\to& \cdots \\ && && &{}^{\lambda}\swarrow& \\ \cdots &\to& 0 &\stackrel{\partial}{\to}& I &\stackrel{\partial}{\to}& 0 &\to& \cdots } \,.$$

Comparing with the general definition of cocycles and coboudnaries from above, one confirms that

• the cocycles are the chain maps

  $$V_\bullet \to I[n]_\bullet$$

• the coboundaries are the chain homotopies between these chain maps.

• the coboundaries of coboundaries are the second order chain homotopies between these chain homotopies.

• etc.

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)

Review of basics

• Charles Weibel, Section 1.1: An Introduction to Homological Algebra

• Michael Hopkins (notes by Akhil Mathew), algebraic topology – Lectures (pdf)