A morphism in a category is called a monic morphism, or monomorphism, if it can be cancelled from the left -- for any object and any morphisms we have if and only if .

A morphism in a category is called a split monomorphism if there exists a morphism such that . Note that every split monomorphism is a monomorphism; if is a split monomorphism and , then one has . By associativity, ; by definition of split monomorphism, ; by definition of identity, , so is a monomorphism. Split monomorphisms are also known as sections and coretractions.

The notion of epimorphism is dual to that of monomorphism. An epimorphism of a category is a monomorphism of the dual category and vice versa.

A monomorphism in the category of sets is simply a one-to-one function. Moreover, in the category of sets all monomorphisms are split monomorphisms.