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regular category
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(Definition)
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A category
 is called a regular category if
- every morphism has a kernel pair,
- every kernel pair has a coequalizer, and
- the pullback of every regular epimorphism along any morphism exists and is again regular. This means the following: if
 is a regular epimorphism, and  is any morphism, then the pullback diagram below
exists, and  is again a regular epimorphism.
Some examples of regular categories are: any abelian category, the category of sets, and the category of groups. On the other hand, the category of topological spaces and the category of small categories are not regular.
Remarks.
- If a category
 is finitely complete, it can be shown that
 is regular iff the strong epimorphisms are stable under pullbacks, and every morphism has a mono-strong-epi factorization: for every morphisms  , we have
 where  is a monomorphism and  is a strong epimorphism.
- Regular categories are generalizations of abelian categories, so that the exactness conditions can be defined without the requirement that the categories be additive. More precisely, in a regular category
 , we define an exact sequence, or exact fork, to be a 6-tuple
 where
such that  is the kernel pair of  and  is the coequalizer of  and  .  is the coequalizer portion of the exact sequence, and  is the kernel pair portion of the exact sequence.
One of the first consequences of the above definition is: every regular epimorphism in a regular category is the coequalizer portion of an exact sequence.
The main result, however, is that, in an abelian category, the two notions of the exactness coincide in the following sense:
 is exact precisely when
is a short exact sequence.
- 1
- F. Borceux Categories and Structures, Handbook of Categorical Algebra II, Cambridge University Press, Cambridge (1994)
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"regular category" is owned by CWoo.
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(view preamble)
Cross-references: short exact sequence, consequences, objects, additive, monomorphism, stable, strong epimorphisms, iff, finitely complete, category of small categories, topological spaces, groups, category of sets, abelian category, regular, regular epimorphism, pullback, coequalizer, kernel pair, morphism, category
There are 10 references to this entry.
This is version 6 of regular category, born on 2008-10-03, modified 2008-11-02.
Object id is 11129, canonical name is RegularCategory.
Accessed 230 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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![$\displaystyle \xymatrix@+=1.5cm{D \ar[r]^h \ar[d] & C \ar[d]^g \\ A \ar[r]_f & B} $](http://images.planetmath.org:8080/cache/objects/11129/l2h/img4.png "$\displaystyle \xymatrix@+=1.5cm{D \ar[r]^h \ar[d] & C \ar[d]^g \\ A \ar[r]_f & B} $")
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![$\displaystyle \xymatrix@+=2cm{0 \ar[r] & A \ar[r]^-{f \choose g} & B\oplus B \ar[r]^-{(h \enspace -h)} & C \ar[r] & 0}$](http://images.planetmath.org:8080/cache/objects/11129/l2h/img26.png "$\displaystyle \xymatrix@+=2cm{0 \ar[r] & A \ar[r]^-{f \choose g} & B\oplus B \ar[r]^-{(h \enspace -h)} & C \ar[r] & 0}$")