Contents

Idea

In the context of homological algebra, for $V_{•}\in {\mathrm{Ch}}_{•}(𝒜)$ 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 $V_{•}$.

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

Basic examples are the singular homology and singular cohomology of a tological 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 $𝒜$ 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 ${\mathrm{Ch}}_{•}(𝒜)$ for the category of chain complexes in $𝒜$. Write ${\mathrm{Ch}}^{•}(𝒜)$ for the category of cochain complexes in $𝒜$.

We label differentials in a chain complex as follows:

Properties

Functoriality

Respect for direct sums and filtered colimits

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

Examples

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

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

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

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/$\mathrm{\infty }$-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

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 $X_{•}$, $A_{•}$ and $B_{•}$ we obtain

• the $\mathrm{\infty }$-groupoid

  $$H_{\mathrm{\infty }\mathrm{Grpd}}(X_{•},A_{•})$$\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 ${\pi }_{0}H(X_{•},A_{•})$ are the cohomology classes

• the $\mathrm{\infty }$-groupoids

  $$H_{\mathrm{\infty }\mathrm{Grpd}}(B_{•},X_{•})$$\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 ${\pi }_{0}H(B_{•},X_{•})$ are the homotopy classes

Chain homology as homotopy

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

A cycle $c:B^n{k}_{•}\to V_{•}$ is a chain map

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 $\lambda :c\to C\prime$ is a chain homotopy

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

(…)

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 $V^{•}$ a cochain complex let

be the corresponding dual chain complex. Let

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

has

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

  $$\begin{array}{ccccccccc}\cdots & \to & V_{n+1}& \stackrel{\partial }{\to }& V_n& \stackrel{\partial }{\to }& V_{n-1}& \to & \cdots \\ & & {\downarrow }^{c_{n+1}}& & {\downarrow }^{c_n}& & {\downarrow }^{c_{n-1}}\\ \cdots & \to & 0& \stackrel{\partial }{\to }& I& \stackrel{\partial }{\to }& 0& \to & \cdots \end{array}.$$\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 $V^{•}=[V_{•},I]$.

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

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

  $$V_{•}\to I[n{]}_{•}$$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

Basics are for instance in section 1.1 of

• Charles Weibel, An Introduction to Homological Algebra

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