Morphism - Some Specific Morphisms

Some Specific Morphisms

• Monomorphism: f : X → Y is called a monomorphism if f ∘ g₁ = f ∘ g₂ implies g₁ = g₂ for all morphisms g₁, g₂ : Z → X. It is also called a mono or a monic.
  • The morphism f has a left inverse if there is a morphism g:Y → X such that g ∘ f = id_X. The left inverse g is also called a retraction of f. Morphisms with left inverses are always monomorphisms, but the converse is not always true in every category; a monomorphism may fail to have a left-inverse.
  • A split monomorphism h : X → Y is a monomorphism having a left inverse g : Y → X, so that g ∘ h = id_X. Thus h ∘ g : Y → Y is idempotent, so that (h ∘ g)2 = h ∘ g.
  • In concrete categories, a function that has a left inverse is injective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.
• Epimorphism: Dually, f : X → Y is called an epimorphism if g₁ ∘ f = g₂ ∘ f implies g₁ = g₂ for all morphisms g₁, g₂ : Y → Z. It is also called an epi or an epic.
  • The morphism f has a right-inverse if there is a morphism g : Y → X such that f ∘ g = id_Y. The right inverse g is also called a section of f. Morphisms having a right inverse are always epimorphisms, but the converse is not always true in every category, as an epimorphism may fail to have a right inverse.
  • A split epimorphism is an epimorphism having a right inverse. Note that if a monomorphism f splits with left-inverse g, then g is a split epimorphism with right-inverse f.
  • In concrete categories, a function that has a right inverse is surjective. Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, every surjection has a section, a result equivalent to the axiom of choice.
• A bimorphism is a morphism that is both an epimorphism and a monomorphism.
• Isomorphism: f : X → Y is called an isomorphism if there exists a morphism g : Y → X such that f ∘ g = id_Y and g ∘ f = id_X. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent. Note that while every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of commutative rings the inclusion Z → Q is a bimorphism, which is not an isomorphism. However, any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism. A category, such as Set, in which every bimorphism is an isomorphism is known as a balanced category.
• Endomorphism: f : X → X is an endomorphism of X. A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h ∘ g with g ∘ h = id. In particular, the Karoubi envelope of a category splits every idempotent morphism.
• An automorphism is a morphism that is both an endomorphism and an isomorphism.