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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/co-concept "Epic" go to Monic and epic morphisms


An arrow, [ilmath]B\mathop{\longrightarrow}^mA[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath] is monic if[1]:

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

This can be stated in a less nasty-looking way as follows:

  • If for each pair [ilmath]X\mathop{\longrightarrow}^{f,\ g}B[/ilmath] of arrows in [ilmath]\mathscr{C} [/ilmath]:
    [ilmath]\xymatrix{ X \ar@<.6ex>[r]^f \ar@<-0.55ex>[r]_g & B \ar[r]^m & A}[/ilmath]
    • this diagram commutes then [ilmath]f=g[/ilmath], [ilmath]f[/ilmath] and [ilmath]g[/ilmath] are the same arrow.

See also


  1. An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition