---
tag: 859F0TFVA3
kind: concept
name: Morphism
msc: [18A05, 18A20]
sources:
  - url: https://en.wikipedia.org/wiki/Morphism
same_as:
  wikidata: Q1948412
  wikipedia: Morphism
---

## Summary

A morphism is an arrow $f \colon A \to B$ of a [[category|A2DE66937D]], from its domain $A$ to its codomain $B$.
Morphisms abstract structure-preserving maps such as [[group homomorphisms|0JYTG5K9Y4]], [[linear maps|0NRYHJPQG2]] and [[continuous maps|0VGFHTM75Y]].
Isomorphisms, endomorphisms and automorphisms are special morphisms defined only through domains, codomains and composition.
Much of [[category theory|Q1RZT7DV0F]] consists of describing objects by the morphisms into them and out of them.

## Explanations

### L1 human

A morphism is an arrow from one object to another that can be chained with other arrows.
In most examples it is a map that respects the structure, but a category does not need to say what the arrow "does" to elements.
An isomorphism is an arrow that can be undone by another arrow.
For instance, in the category of vector spaces over $\R$, an invertible $2 \times 2$ matrix gives an isomorphism $\R^2 \to \R^2$.
A matrix with determinant $0$ gives a morphism that is not an isomorphism.