nLab
homological category

Homological category

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Regular and exact categories

∞-ary regular and exact categories

regularity

exactness

Homological category

Definition

A category CC is homological if it is

  1. pointed

  2. regular

  3. protomodular.

Properties

Many of the standard results of classical homological algebra in abelian categories extend to homological categories:

A homological category which is Barr-exact and has finite coproducts is semiabelian.

Examples

Example

The category Grp of all groups (including non-abelian groups) is homological. Namely it is

  1. regular, by this example,

  2. pointed protomodular by this example.

References

Last revised on September 22, 2021 at 05:31:52. See the history of this page for a list of all contributions to it.