Normal morphism

<h2 id="definition">Definition</h2>
A monomorphism is normal if it is the <a href="/facts/Kernel_(category_theory)/Tsn0uLfc">kernel</a> of some morphism, and an epimorphism is conormal if it is the <a href="/facts/Cokernel_(category_theory)/vYFQRPoh">cokernel</a> 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.

<h2 id="examples">Examples</h2>
In the <a href="/facts/Category_of_groups/sxSd9Kbc">category of groups</a>, a monomorphism f from H to G is normal <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> its image is a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> of G. In particular, if H is a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of G, then the <a href="/facts/Inclusion_map/VUFNdaDH">inclusion map</a> 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 conormal (since it is the cokernel of its own kernel), so this category is conormal.
In an <a href="/facts/Abelian_category/4WN473Yc">abelian category</a>, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel.
Thus, abelian categories are always binormal.
The category of <a href="/facts/Abelian_group/FfWwmx4R">abelian groups</a> is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.

<ul><li>Section I.14 Mitchell, Barry (1965). Theory of categories. Pure and applied mathematics. Vol. 17. Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-124-99250-4. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0202787">0202787</a>.</li></ul>

Normal morphism open-in-new

Normal morphism