---
tag: AQET9YHH6F
kind: concept
name: Zero object
msc: [18A30, 18A20]
specializes: [EK85VSSRMR]
sources:
  - url: https://en.wikipedia.org/wiki/Zero_object_(algebra)
---

## Summary

A zero object of a [[category|A2DE66937D]] is an object that is both [[initial and terminal|EK85VSSRMR]].
Categories with a zero object, such as groups, modules and pointed sets, have a distinguished zero morphism between any two objects.
This is the starting point for kernels, cokernels and exact sequences in homological algebra.

## Explanations

### L1 human

A zero object has exactly one arrow to every object and exactly one arrow from every object.
The [[trivial group|0BNPC9850Y]] $\{e\}$ is a zero object among groups.
There is one homomorphism from it to any group, sending $e$ to the identity.
There is one homomorphism to it from any group, sending everything to $e$.
The zero vector space plays the same role among vector spaces.
Among sets there is no zero object: the empty set is initial but not terminal, and a one-point set is terminal but not initial.