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# epi

A morphism $f:A\to B$ in a category $\mathcal{C}$ is called epi if for any object $C$ and any morphisms $g_{1},g_{2}:B\to C$, if $g_{1}f=g_{2}f$ then $g_{1}=g_{2}$. In other words, any diagram

$\xymatrix{A\ar[r]^{f}&B\ar[r]^{{g_{1}}}&C}=\xymatrix{A\ar[r]^{f}&B\ar[r]^{{g_{% 2}}}&C}$

reduces to the diagram

$\xymatrix{B\ar[r]^{{g_{1}}}&C}=\xymatrix{B\ar[r]^{{g_{2}}}&C}.$ |

An *epimorphism* is just an epi morphism, and epi is also known as *right cancellable*, *epimorphic*, or simply *epic*.

Remarks.

1. If $\mathcal{C}$ is an abelian category, then an epi has the property that $gf=0$ implies $g=0$ (surely, since $gf=0=0f$, and the result follows).

2. Epi is the generalization of a function being onto. In some categories where surjections are well-defined (such as sets and groups), epi is the same as being onto.

3. The dual notion of epi is that of monic.

## Mathematics Subject Classification

18A20*no label found*18A05

*no label found*

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