Chain complex of groups

This article defines the notion of group object in the category of based-set chain complexs|View other types of group objects

Definition

A chain complex of groups is defined as the following data:

• A collection $G_n$ of groups, where $n$ varies over the integers
• A collection $d_n:G_n\to G_{n-1}$ of group homomorphisms

such that for any $n$:

$d_{n-1}\circ d_n=0$

Terminology

• If the image of $d_n$ equals the kernel of $d_{n-1}$, then the chain complex is said to be exact at $G_{n-1}$
• If the image of $d_n$ is a normal subgroup of the kernel of ,math>d_{n-1}</math>, the quotient group is termed the $(n-1{)}^{th}$ homology group of the chain complex.