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types of morphisms (Definition)
Definition 1   A morphism $ f:A\to B$ is a monomorphism, if for any two morphisms $ g,h:B\to C$ the equality $ f\circ g=f\circ h$ implies $ h=g$.

Dual notion: Morphism $ f:A\to B$ is an epimorphism, if for any two morphisms $ g,h:B\to C$ the equality $ g\circ f=h\circ f$ implies $ h=g$.

A morphism $ f$ is a bimorphism, if it is monomorphism and epimorphism at the same time.

Definition 2   A morphism $ f:A\to B$ is called retraction if there exists a morphism $ g:B\to A$ such that $ f\circ g=id_B$.

Retractions are sometimes called split epimorphisms.

Dual notion: a morphism $ f:A\to B$ is a section (or coretraction or split monomorphism) if there exists a morphism $ g:B\to A$ such that $ g\circ f=id_A$.

A morphism $ f:A\to B$ is an isomorphism if it is a retraction and section at the same time.

Bimorphism and isomorphism are examples of self-dual properties. The condition that $ f$ is isomorphism is equivalent to the existence of a morphism $ g$ with $ f\circ g=id_B$ and $ g\circ f=id_A$ (for the proof see properties of monomorphisms and epimorphisms).

Definition 3   If $ f$ is an isomorphism then the morphism $ g:B\to A$ such that $ f\circ g=id_B$ and $ g\circ f=id_A$ is called inverse morphism of $ f$ and denoted by $ f^{-1}$.
Definition 4   If there exists an isomorphism $ f:A\to B$ we say that the objects $ A$ and $ B$ are isomorphic, denoted by $ A\cong B$.



"types of morphisms" is owned by kompik.
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Also defines:  monomorphism, epimorphism, bimorphism, retraction, section, coretraction, isomorphism, inverse morphism, split monomorphism, split epimorphism

Attachments:
properties of monomorphisms and epimorphisms (Theorem) by kompik
extremal monomorphism (Definition) by kompik
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Cross-references: isomorphic, objects, inverse, properties of monomorphisms and epimorphisms, equivalent, properties, self-dual, implies, equality, morphism
There are 27 references to this entry.

This is version 7 of types of morphisms, born on 2006-06-30, modified 2007-02-20.
Object id is 8114, canonical name is TypesOfMorphisms.
Accessed 4526 times total.

Classification:
AMS MSC: 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
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Types of morphisms by kompik on 2006-06-30 09:17:33
I've added this entry despite the fact that some of these notions are already defined somewhere in PM. In my opinion it's good to have these definitions gathered in one place.

They can be found e.g. in
http://planetmath.org/encyclopedia/AbelianCategory.html
http://planetmath.org/encyclopedia/Epi.html
http://planetmath.org/encyclopedia/Monic.html

I've seen that epimorphism or monomorphism often redirects to
http://planetmath.org/encyclopedia/TypesOfHomomorphisms.html
where these notions are defined from the point of view of Universal Algebra, not category theory.

Regular mono/epimorphisms are defined in
http://planetmath.org/encyclopedia/Equalizer.html

I'll add extremal mono/epimorphisms in a separate entry.
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