Category

Morphism

A morphism is an element of some category. 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.

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 functors and natural transformations.

Mnemonic for types of morphism → MILESR

• Monic
• Injective
• Left-cancellable
• Epic
• Surjective
• Right-cancellable

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tidy | simplify | mnemonic