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normal category
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(Definition)
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A monomorphism is a category is said to be normal if it is a kernel (of a morphism). A subobject of an object is normal if any (and hence all) of its representing monomorphisms is normal.
For example, in Grp, the category of groups, the inclusion of a subgroup
 into  is normal iff  is a normal subgroup of  .
A category is said to be normal if every monic is a kernel. Equivalently, a normal category is a category in which every subobject of every object is normal.
Dually, an epimorphism is conormal if it is a cokernel (of a morphism). A quotient object of an object is conormal if any (and hence all) of its representing epimorphisms is conormal. A category is said to be conormal if every epimorphism is conormal.
The category
 of abelian groups, and more generally, any abelian category, is normal and conormal.
- 1
- C. Faith Algebra: Rings, Modules, and Categories I, Springer-Verlag, New York (1973)
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"normal category" is owned by CWoo.
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(view preamble | get metadata)
| Other names: |
normal monic, conormal epi |
| Also defines: |
normal, normal monomorphism, normal subobject, conormal, conormal epimorphism, conormal category, conormal quotient object |
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Cross-references: abelian category, abelian groups, quotient object, cokernel, epimorphism, monic, iff, subgroup, inclusion, groups, object, subobject, morphism, kernel, category, monomorphism
There are 4 references to this entry.
This is version 5 of normal category, born on 2008-09-01, modified 2008-09-22.
Object id is 10975, canonical name is NormalCategory.
Accessed 606 times total.
Classification:
| AMS MSC: | 18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories) |
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Pending Errata and Addenda
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