[[Category]]
# Morphism

A **morphism** is an element of some [[category]]. #m/def/cat 
Typically, it represents some kind of structure-preserving map between sets,
or a more abstract relationship between objects.
Morphisms have the important property that they may be composed to produce new morphisms.

## Classification

In the following, arrows signify implication.

```mermaid
graph TD;
    Automorphism:::internal-link
    Isomorphism:::internal-link
    Endomorphism:::internal-link
    SMonomorphism[Split monomorphism]:::internal-link
    SEpimorphism[Split epimorphism]:::internal-link
    RMonomorphism[Regular monomorphism]:::internal-link
    REpimorphism[Regular epimorphism]:::internal-link
    Monomorphism:::internal-link
    Epimorphism:::internal-link
    Morphism:::internal-link

    Automorphism==>Endomorphism==>Morphism
    Automorphism==>Isomorphism
    Isomorphism==>SMonomorphism==>RMonomorphism==>Monomorphism==>Morphism
    Isomorphism==>SEpimorphism==>REpimorphism==>Epimorphism==>Morphism
```


The same prefixes are used for specific morphisms, including [[Functor|functors]] and [[Natural transformation|natural transformations]].

Mnemonic for types of morphism
→
`MILESR`
- **M**onic
- **I**njective
- **L**eft-cancellable 
- **E**pic
- **S**urjective
- **R**ight-cancellable
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