Types of category arrows

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There are many kinds of arrows in a category, here are some common terms compared, and a diagram showing how they relate. [ilmath]\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] }[/ilmath]

Monic & Epic

Monic captures the idea of an injection

[ilmath]\xymatrix{X \ar@<-.5ex>[r]_g \ar@<.5ex>[r]^f & B \ar[r]^m & A} [/ilmath]
(warning: non-commutative diagram)
[ilmath]m[/ilmath] is monic if:
  • [ilmath]\forall X\in\text{Ob}(\mathscr{C})\forall f,g\in\text{Arw}_\mathscr{C}(X,B)[(m\circ f=m\circ g)\implies f=g][/ilmath]

And epic, which captures the idea of a surjection

[ilmath]\xymatrix{A \ar[r]^e & B \ar@<-.5ex>[r]_g \ar@<.5ex>[r]^f & X} [/ilmath]
(warning: non-commutative diagram)
[ilmath]m[/ilmath] is epic if:
  • [ilmath]\forall X\in\text{Ob}(\mathscr{C})\forall f,g\in\text{Arw}_\mathscr{C}(B,X)[(f\circ e=g\circ e)\implies f=g][/ilmath]