You are here
Homeregular category
Primary tabs
regular category
A category $\mathcal{C}$ is called a regular category if
1. every morphism has a kernel pair,
2. every kernel pair has a coequalizer, and
3. the pullback of every regular epimorphism along any morphism exists and is again regular. This means the following: if $f:A\to B$ is a regular epimorphism, and $g:C\to B$ is any morphism, then the pullback diagram below
$\xymatrix@+=1.5cm{D\ar[r]^{h}\ar[d]&C\ar[d]^{g}\\ A\ar[r]_{f}&B}$ exists, and $h$ 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 $\mathcal{C}$ is finitely complete, it can be shown that $\mathcal{C}$ is regular iff the strong epimorphisms are stable under pullbacks, and every morphism has a monostrongepi factorization: for every morphisms $f$, we have $f=g\circ h$ where $g$ is a monomorphism and $h$ 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 $\mathcal{C}$, we define an exact sequence, or exact fork, to be a 6tuple $(A,B,C,f,g,h)$ where

$A,B,C$ are objects

$f,g:A\to B$ and $h:B\to C$ are morphisms: $\xymatrix@+=2cm{A\ar@<0.5ex>[r]^{f}\ar@<0.5ex>[r]_{g}&B\ar[r]^{h}&C}$
such that $(f,g)$ is the kernel pair of $h$ and $h$ is the coequalizer of $f$ and $g$. $h$ is the coequalizer portion of the exact sequence, and $(f,g)$ 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: $(A,B,C,f,g,h)$ is exact precisely when
$\xymatrix@+=2cm{0\ar[r]&A\ar[r]^{}{f\choose g}&B\oplus B\ar[r]^{}{(h\enspace% h)}&C\ar[r]&0}$ is a short exact sequence.

References
 1 F. Borceux Categories and Structures, Handbook of Categorical Algebra II, Cambridge University Press, Cambridge (1994)
Mathematics Subject Classification
18E10 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections