Epic

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Just a quick thing, needs checking and adding any missing things
Note: to see this concept discussed with its dual/twin/co-concept "Monic" go to Monic and epic morphisms

Definition

An arrow, [ilmath]A\mathop{\longrightarrow}^eB[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath] is epic if[1]:

• $\forall X\in\text{Ob}(\mathscr{C})\ \forall f,g\in\text{Hom}_\mathscr{C}(B,X)[(f\circ e=g\circ e)\implies f=g]$

This can be stated in a less nasty-looking 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.