morphism

A morphism is the categorical word for an arrow — the abstraction of “a structure-respecting map from one thing to another.” In familiar categories morphisms really are functions of some kind (homomorphisms, continuous maps, linear maps), but the category-theoretic stance is to treat them as primitive: a morphism need not be a function on underlying elements at all. What is fixed is that each morphism f has a source object (domain) and a target object (codomain), written f : A -> B.

Several adjectives classify morphisms purely by how they compose, not by anything internal. A morphism f : A -> B is a monomorphism if it is left-cancellable: f ∘ g = f ∘ h forces g = h. It is an epimorphism if it is right-cancellable: g ∘ f = h ∘ f forces g = h. It is an isomorphism if it has a two-sided inverse g with g ∘ f = id_A and f ∘ g = id_B. An endomorphism is a morphism A -> A, and an automorphism is an invertible endomorphism.

A common trap: in Set, monomorphism means injective and epimorphism means surjective, and a map that is both is an isomorphism. But this last implication fails in general categories. In the category of rings, the inclusion Z -> Q is both monic and epic yet is not an isomorphism. So “mono and epi” is strictly weaker than “iso” outside nice settings, and the three adjectives must be kept conceptually distinct.