# Tag Info

A chain complex $(A_{\bullet}, d_{\bullet})$ is a sequence $(A_n)_{-\infty}^{\infty}$ of abelian groups (or modules) and group (module) homomorphisms $d_n : A_n \to A_{n-1}$ such that $d_{n-1}\circ d_n = 0$. This data can be represented as follows:

$$\cdots \xrightarrow{d_{n+1}} A_n \xrightarrow{d_n} A_{n-1} \xrightarrow{d_{n-1}} \cdots$$

The homology of a chain complex is the sequence of abelian groups

$$H_n = \frac{\ker d_n}{\operatorname{im}d_{n+1}}.$$

Dually, a cochain complex is a sequence $(A_{\bullet}, d_{\bullet})$ of abelian groups where $d_n : A_n \to A_{n+1}$.

There are many common types of (co)homology including simplicial (co)homology, singular (co)homology, and group (co)homology. A more extensive list can be found here.

Simplicial homology and singular homology are examples of homology theories attached to a topological space. The Eilenberg-Steenrod axioms are a collection of properties that such homology theories share.

For the more abstract aspects of (co)homology theory, the tag may be more appropriate.