# nLab homological category

Homological category

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

### Theorems

#### Regular and exact categories

∞-ary regular and exact categories

regularity

exactness

# Homological category

## Definition

A category $C$ is homological if it is

## 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.