Glossary of Category Theory - Morphisms

Morphisms

A morphism f in a category is called:

• an epimorphism provided that whenever . In other words, f is the dual of a monomorphism.
• an identity provided that f maps an object A to A and for any morphisms g with domain A and h with codomain A, and .
• an inverse to a morphism g if is defined and is equal to the identity morphism on the codomain of g, and is defined and equal to the identity morphism on the domain of g. The inverse of g is unique and is denoted by g−1. f is a left inverse to g if is defined and is equal to the identity morphism on the domain of g, and similarly for a right inverse.
• an isomorphism provided that there exists an inverse of f.
• a monomorphism (also called monic) provided that whenever ; e.g., an injection in Set. In other words, f is the dual of an epimorphism.
• a retraction if it has a right inverse.
• a coretraction if it has a left inverse.