Epic
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Just a quick thing, needs checking and adding any missing things
 Note: to see this concept discussed with its dual/twin/coconcept "Monic" go to Monic and epic morphisms
Contents
Definition
An arrow, [ilmath]A\mathop{\longrightarrow}^eB[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath] is epic if^{[1]}:
 [math]\forall X\in\text{Ob}(\mathscr{C})\ \forall f,g\in\text{Hom}_\mathscr{C}(B,X)[(f\circ e=g\circ e)\implies f=g][/math]
This can be stated in a less nastylooking way as follows:
 If for each pair [ilmath]B\mathop{\longrightarrow}^{f,\ g}X[/ilmath] of arrows in [ilmath]\mathscr{C} [/ilmath]:
 [ilmath]\xymatrix{ A \ar[r]^e & B \ar@<.6ex>[r]^f \ar@<0.55ex>[r]_g & X}[/ilmath]
 this diagram commutes then [ilmath]f=g[/ilmath], [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are the same arrow.
See also
References
