Monic

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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

Definition

An arrow, $B\mathop{\longrightarrow}^mA$ in a category $\mathscr{C}$ is monic if:

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

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

• If for each pair $X\mathop{\longrightarrow}^{f,\ g}B$ of arrows in $\mathscr{C}$:

  $\xymatrix{ X \ar@<.6ex>[r]^f \ar@<-0.55ex>[r]_g & B \ar[r]^m & A}$

  • this diagram commutes then $f=g$, $f$ and $g$ are the same arrow.

See also

• Epic
• Monic and epic morphisms
• Types of category arrows