---
tag: H8W38933NC
kind: concept
name: Cokernel
msc: [18A30, "18E05"]
sources:
  - url: https://en.wikipedia.org/wiki/Cokernel
same_as:
  wikidata: Q2156511
  wikipedia: Cokernel
---

## Summary

In a category with zero morphisms, the cokernel of a morphism $f$ is the universal morphism $q$ with $q \circ f = 0$.
It is the [[coequalizer|DVSTWGVCSZ]] of $f$ and the zero morphism, and the notion dual to the [[kernel|S417YNEJ6M]].
For modules, the cokernel of $f \colon X \to Y$ is the quotient map $Y \to Y / f(X)$.
The cokernel measures how far a morphism is from being surjective.

## Explanations

### L1 human

The image of a linear map is the part of the target that the map reaches.
The cokernel is what is left when one divides the target by the image: it records the directions the map misses.
A map onto its target has zero cokernel.
For multiplication by $n$ on $\Z$, the image is $n\Z$ and the cokernel is $\Z / n\Z$.