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types of morphisms
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(Definition)
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Definition 1 A morphism $\Map fAB$ is a monomorphism, if for any two morphisms $\Map{g,h}CA$ the equality $f\circ g=f\circ h$ implies $h=g$ .
Dual notion: Morphism $\Map fAB$ is an epimorphism, if for any two morphisms $\Map{g,h}BC$ 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. Also the names epimorphic extension and epimorphic monomorphism are used.
Definition 2 A morphism $\Map fAB$ is called retraction if there exists a morphism $\Map gBA$ such that $f\circ g=id_B$ .
Retractions are sometimes called split epimorphisms.
Dual notion: a morphism $\Map fAB$ is a section (or coretraction or split monomorphism) if there exists a morphism $\Map gBA$ such that $g\circ f=id_A$ .
A morphism $\Map fAB$ 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 $\Map gBA$ such that $f\circ g=id_B$ and $g\circ f=id_A$ is called inverse morphism of $f$ and denoted by $\inv f$ .
Definition 4 If there exists an isomorphism $\Map fAB$ we say that the objects $A$ and $B$ are isomorphic, denoted by $A\cong B$ .
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"types of morphisms" is owned by kompik.
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See Also: types of homomorphisms, sections and retractions
| Also defines: |
monomorphism, epimorphism, bimorphism, retraction, section, coretraction, isomorphism, inverse morphism, split monomorphism, split epimorphism, epimorphic extension, epimorphic monomorphism |
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Cross-references: isomorphic, objects, inverse, properties of monomorphisms and epimorphisms, proof, equivalent, properties, self-dual, implies, equality, morphism
There are 65 references to this entry.
This is version 13 of types of morphisms, born on 2006-06-30, modified 2008-10-08.
Object id is 8114, canonical name is TypesOfMorphisms.
Accessed 10244 times total.
Classification:
| AMS MSC: | 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations) |
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Pending Errata and Addenda
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