Types of category arrows

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]