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In category theory and its applications to mathematics, a **normal monomorphism** or **conormal epimorphism** is a particularly well-behaved type of morphism.
A **normal category** is a category in which every monomorphism is normal. A **conormal category** is one in which every epimorphism is conormal.

A monomorphism is **normal** if it is the kernel of some morphism, and an epimorphism is **conormal** if it is the cokernel of some morphism.

A category **C** is **binormal** if it's both normal and conormal.
But note that some authors will use the word "normal" only to indicate that **C** is binormal.

In the category of groups, a monomorphism *f* from *H* to *G* is normal if and only if its image is a normal subgroup of *G*. In particular, if *H* is a subgroup of *G*, then the inclusion map *i* from *H* to *G* is a monomorphism, and will be normal if and only if *H* is a normal subgroup of *G*. In fact, this is the origin of the term "normal" for monomorphisms.

On the other hand, every epimorphism in the category of groups is normal (since it is the cokernel of its own kernel), so this category is conormal.

This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Normal_morphism

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