Bimorphism

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Definition

A bimorphism is a morphism or arrow in a category $\mathscr{C}$ ^[1]:

$\xymatrix{ A \ar[r]^f & B}$

That is

both monic and epic

Warning:A bimorphism need not be an isomorphism, when all bimorphisms in $\mathscr{C}$ are isomorphisms however, we say that $\mathscr{C}$ is balanced

References

Jump up ↑ An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition

v • d • e Category Theory

Overview of the concepts of Category Theory

Key objects	Category, Functor (Covariant, Contravariant), Subcategory	$\xymatrix{ & \text{Arrow} \\ \text{Monic} \ar@{^{(}->}[ur] & & \text{Epic} \ar@{^{(}->}[ul] \\ & \text{Bimorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \\ {\begin{array}{c}\text{Section}\\ \text{(Split monic)} \end{array} } \ar@{^{(}->}[uu] & & {\begin{array}{c}\text{Retraction}\\ \text{(Split epic)} \end{array} } \ar@<-0.75ex>@{^{(}->}[uu] \\ & \text{Isomorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \ar@{^{(}->}[uu] }$

Typical morphism types (see diagram on right)	Arrow (AKA: Morphism), Monic, Epic, Bimorphism, Section (AKA: Split monic), Retraction (AKA: Split epic), Isomorphism

Key objects	$\text{Initial}$ , $\text{Final}$ (compared)

Primitive constructs	Wedge (AKA Cone & Cocone)

Key constructs	Product / Coproduct (Product and coproduct compared), Limit / Colimit, Equaliser / Coequaliser

Important examples	Demonstrating why category arrows are best thought of as arrows and not functions

Trivial category examples	Category induced by a monoid, Category induced by a poset

Common categories	$\mathrm{SET}$ , $\mathrm{Pfn}$ , $\mathrm{GROUP}$