Types of category arrows

Overview

There are many kinds of arrows in a category, here are some common terms compared, and a diagram showing how they relate. $\xymatrix{ & \text{Arrow} \\ \text{Monic} \ar@{^{(}->}[ur] & & \text{Epic} \ar@{^{(}->}[ul] \\ & \text{Bimorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \\ {\begin{array}{c}\text{Section}\\ \text{(Split monic)} \end{array} } \ar@{^{(}->}[uu] & & {\begin{array}{c}\text{Retraction}\\ \text{(Split epic)} \end{array} } \ar@<-0.75ex>@{^{(}->}[uu] \\ & \text{Isomorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \ar@{^{(}->}[uu] }$

Monic & Epic

Monic captures the idea of an injection

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

And epic, which captures the idea of a surjection

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