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Hometypes of morphisms

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# types of morphisms

###### Definition 1.

A morphism $f:A\to B$ is a *monomorphism*, if for any two morphisms $g,h:C\to A$ 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. Also the names *epimorphic extension* and *epimorphic monomorphism* are used.

###### 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 $A\cong B$.

## Mathematics Subject Classification

18A05*no label found*

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## Attached Articles

## Corrections

split monomorphism by rspuzio ✓

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monomorphism not epimorphism by CWoo ✓

bimorphism by jocaps ✓

## Comments

## Types of morphisms

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.

## Re: Types of morphisms

quetion: do it makes sense to speak of a <<homomorphism>> between open sets of complex vector spaces?