---
tag: YF9F9EG6M3
kind: concept
name: Zero morphism
msc: [18A20, 18A05]
specializes: [859F0TFVA3]
sources:
  - url: https://en.wikipedia.org/wiki/Zero_morphism
same_as:
  wikidata: Q2905069
  wikipedia: Zero morphism
---

## Summary

A zero morphism is a distinguished [[morphism|859F0TFVA3]] between two objects that behaves like multiplication by zero.
Composing it with any morphism on either side gives again a zero morphism.
In a category with a [[zero object|AQET9YHH6F]], the zero morphism $X \to Y$ is the composite $X \to 0 \to Y$.
Zero morphisms are needed to define kernels and cokernels, and they make sense of the statement that a composite is zero.

## Explanations

### L1 human

In the category of groups, the zero morphism from $G$ to $H$ is the homomorphism that sends every element of $G$ to the identity of $H$.
It is the "do nothing useful" map, and it exists between any two groups.
If you follow it with any homomorphism, or precede it with any homomorphism, you still send everything to the identity.
For linear maps between vector spaces, the zero morphism is the zero map $v \mapsto 0$.