View source for Normal epimorphism

A morphism having the characteristic property of the natural mapping of a group onto a quotient group or of a ring onto a quotient ring. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674801.png" /> be a [[Category|category]] with zero morphisms. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674802.png" /> is called a normal epimorphism if every morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674803.png" /> for which it always follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674805.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674806.png" />, can be uniquely represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674807.png" />. The [[Cokernel|cokernel]] of any morphism is a normal epimorphism. The converse assertion is false, in general; however, when morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067480/n0674808.png" /> have kernels, then every normal epimorphism is a cokernel. In an [[Abelian category|Abelian category]] every epimorphism is normal. The concept of a normal epimorphism is dual to that of a [[Normal monomorphism|normal monomorphism]].