nLab
homology

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

$$N_{•}:\mathrm{SimpAb}\stackrel{\simeq }{\to }{\mathrm{Ch}}_{+}.$$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 oppposite: 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.

But conceptually it is 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

$$V_{•}=(\cdots \to V_{n+1}\stackrel{{\partial }_n}{\to }{V}_n\stackrel{{\partial }_{n-1}}{\to }{V}_{n-1}\to \cdots )$$V_\bullet = ( \cdots \to V_{n+1} \stackrel{\partial_n}{\to} V_n \stackrel{\partial_{n-1}}{\to} V_{n-1} \to \cdots )

be a chain complex in $C$. For each interger $n\in \mathbb{N}$ this induces the following diagram

$$\begin{array}{ccccccc}& & \mathrm{im}{\delta }_n& & \to & & \ker {\delta }_{n-1}\\ & \nearrow & & \searrow & & \swarrow \\ V_{n+1}& & \stackrel{{\partial }_n}{\to }& & V_n& & \stackrel{{\partial }_{n-1}}{\to }& & V_{n-1}\\ & & & \swarrow & & \searrow & & \nearrow \\ & & \mathrm{coker}{\delta }_n& & \stackrel{}{\to }& & \mathrm{im}{\delta }_{n-1}\end{array}$$\array{ && im \delta_n &&\to&& ker \delta_{n-1} \\ & \nearrow && \searrow && \swarrow \\ V_{n+1} &&\stackrel{\partial_n}{\to}&& V_n &&\stackrel{\partial_{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

$$\begin{array}{rl}\mathrm{im}(\ker {\delta }_{n-1}\to V_n\to \mathrm{coker}\delta )& \simeq \mathrm{coker}(\mathrm{im}{\delta }_n\to \ker {\delta }_{n-1})\\ & \simeq \mathrm{coker}(V_{n+1}\to \ker {\delta }_{n-1})\\ & \simeq \ker (\mathrm{coker}{\delta }_n\to \mathrm{im}{\delta }_{n-1})\\ & \simeq \ker (\mathrm{coker}{\delta }_n\to V_{{\delta }_{n-1}})\end{array}$$\begin{aligned} im(ker \delta_{n-1} \to V_n \to coker \delta) & \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_{\delta_{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 $V_{•}$ is the quotient group

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