types of morphismsTypesOfMorphisms2006-06-30 09:10:572008-10-08 12:43:13Definitionmonomorphismepimorphismbimorphismretractionsectioncoretractionisomorphisminverse morphismsplit monomorphismsplit epimorphismepimorphic extensionepimorphic monomorphism% this is the default PlanetMath preamble. as your knowledge
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& {#6} } }\begin{DEF}
A morphism $\Map fAB$ is a \emph{monomorphism}, if for any two morphisms $\Map{g,h}CA$ the
equality $f\circ g=f\circ h$ implies $h=g$.
\PMlinkname{Dual}{DualityPrinciple} notion: Morphism $\Map fAB$ is an \emph{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 \emph{bimorphism}, if it is monomorphism and epimorphism at the same
time. Also the names \emph{epimorphic extension} and \emph{epimorphic monomorphism} are used.
\end{DEF}
\begin{DEF}
A morphism $\Map fAB$ is called \emph{retraction} if there exists a morphism $\Map gBA$ such
that $f\circ g=id_B$.
Retractions are sometimes called \emph{split epimorphisms}.
Dual notion: a morphism $\Map fAB$ is a \emph{section} (or \emph{coretraction} or \emph{split monomorphism}) if there exists a morphism $\Map gBA$ such that $g\circ f=id_A$.
A morphism $\Map fAB$ is an \emph{\PMlinkname{isomorphism}{Isomorphism2}} if it is a
retraction and section at the same time.
\end{DEF}
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).
\begin{DEF} 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 \emph{inverse} morphism of $f$ and denoted by $\inv f$.
\end{DEF}
\begin{DEF}
If there exists an isomorphism $\Map fAB$ we say that the objects $A$ and $B$ are
\emph{isomorphic}, denoted by $A\cong B$.
\end{DEF}