nLab homology

Of chain complexes

Idea

Under the Dold-Kan correspondence, ∞-groupoids with strict abelian group structure (modeled by Kan complexes that are simplicial abelian groups) are identified with non-negatively graded chain complexes of abelian groups

$$N_\bullet : SimpAb \stackrel{\simeq}{\to} Ch_+ \,.$$

The homology groups of a chain complex of abelian groups are the image under this identification of the homotopy groups of the corresponding ∞-groupoids. More details on this are at chain homology and cohomology.

So at least for the case of chain complexes of abelian groups we have the slogan

homology = homotopy under Dold-Kan correspondence

Of course historically the development of concepts was precisely the opposite: chain homology is an old fundamental concept in homological algebra that is simpler to deal with than simplicial homotopy groups. The computational simplification for chain complexes is what makes the Dold-Kan correspondence useful after all.

Conceptually, however, it can be useful to understand homology as a special kind of homotopy. This is maybe most vivid in the dual picture: cohomology derives its name from that fact that chain homology and cohomology are dual concepts. But later generalizations of cohomology to generalized (Eilenberg-Steenrod) cohomology and further to nonabelian cohomology showed that the restricted notion of homology is an insufficient dual model for cohomology: what cohomology is really dual to is the more general concept of homotopy. More on this is at cohomotopy and Eckmann-Hilton duality.

Definition

The category of abelian groups is in particular an abelian category. We can define chain complexes and their homology in any abelian category $C$.

Let $C$ be an abelian category and let

$$V_\bullet = ( \cdots \to V_{n+1} \stackrel{\delta_n}{\to} V_n \stackrel{\delta_{n-1}}{\to} V_{n-1} \to \cdots )$$

be a chain complex in $C$. For each integer $n \in \mathbb{N}$ this induces the following diagram of kernels, cokernels and images

$$\array{ && im \delta_n &&\to&& ker \delta_{n-1} \\ & \nearrow && \searrow && \swarrow \\ V_{n+1} &&\stackrel{\delta_n}{\to}&& V_n &&\stackrel{\delta_{n-1}}{\to}&& V_{n-1} \\ & && \swarrow && \searrow && \nearrow \\ && coker \delta_n &&\stackrel{}{\to}&& im \delta_{n-1} }$$

the homology $H_n(V)$ of $V$ in degree $n$ is the object

$$\begin{aligned} im(ker \delta_{n-1} \to V_n \to coker \delta_{n}) & \simeq coker(im \delta_n \to ker \delta_{n-1}) \\ & \simeq coker(V_{n+1} \to ker \delta_{n-1}) \\ & \simeq ker(coker \delta_n \to im \delta_{n-1}) \\ & \simeq ker(coker \delta_n \to V_{n-1}) \end{aligned}$$

• If $H_n(V) \simeq 0$ then one says that the complex $V$ is exact in degree $n$.

Examples

In the special case that $C$ is the category of abelian groups, or of vector spaces, this definition reduces to the more familiar simpler statement:

the $n$-th homology group of the chain complex $V_\bullet$ is the quotient group

$$H_n(V) = ker(\partial_n) / im(\partial_{n+1}) \,.$$

Generalized homology

By the Brown representability theorem every spectrum $A$ induces a generalized (Eilenberg-Steenrod) cohomology theory, and dually a generalized homology theory.

For $X$ a topological space and $A$ a spectrum, the generalized homology of spectrum of $X$ with coefficients in $A$ is

$$X \wedge A := \Sigma^\infty(X)\wedge A \,,$$

where on the right we have the smash product of spectra with the suspension spectrum of $X$ and on the left we abbreviate this to the (∞,1)-tensoring of Spec over Top.

The corresponding homology groups are the homotopy groups of this spectrum:

$$E_n(X,A) := \pi_n(X \wedge A) := [\Sigma^n \mathbb{S}, X \wedge A ] \,.$$

where $\mathbb{S}$ is the sphere spectrum. For more see generalized homology.

Examples

• chain homology

  • relative homology

• simplicial homology

• singular homology

  • cellular homology

• generalized homology

• factorization homology

• Kan-Thurston Theorem

Related concepts

• homology of loop spaces

• effective homology

• persistent homology

The relation between homology, cohomology and homotopy:

	homotopy	cohomology	homology
	$[S^n,-]$	$[-,A]$	$(-) \otimes A$
category theory	covariant hom	contravariant hom	tensor product
homological algebra	Ext	Ext	Tor
enriched category theory	end	end	coend
homotopy theory	derived hom space $\mathbb{R}Hom(S^n,-)$	cocycles $\mathbb{R}Hom(-,A)$	derived tensor product $(-) \otimes^{\mathbb{L}} A$

The ingredients of homology and cohomology:

$H_n = Z_n/B_n$	(chain-)homology	(cochain-)cohomology	$H^n = Z^n/B^n$
$C_n$	chain	cochain	$C^n$
$Z_n \subset C_n$	cycle	cocycle	$Z^n \subset C^n$
$B_n \subset C_n$	boundary	coboundary	$B^n \subset C^n$

References

• Peter Hilton (ed.) Category Theory, Homology Theory and Their Applications,

  vol 1: Lecture Notes in Mathematics 86, Springer (1969) [doi:10.1007/BFb0079380]

  vol 2: Lecture Notes in Mathematics 92, Springer (1969) [doi:10.1007/BFb0080761]

  vol 3: Lecture Notes in Mathematics 99, Springer (1969) [doi:10.1007/BFb0081959]