Regular category

In category theory, a regular category is a category with finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain "exactness" conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of "images", without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.
Definition.
A category "C" is called regular if it satisfies the following three properties:
 
 
Examples.
Examples of regular categories include:
The following categories are "not" regular:
Epi-mono factorization.
In a regular category, the regular-epimorphisms and the monomorphisms form a factorization system. Every morphism "f:X→Y" can be factorized into a regular epimorphism "e:X→E" followed by a monomorphism "m:E→Y", so that "f=me". The factorization is unique in the sense that if "e':X→E' "is another regular epimorphism and "m':E'→Y" is another monomorphism such that "f=m'e, then there exists an isomorphism "h:E→E' " such that "he=e' "and "m'h=m". The monomorphism "m" is called the image"' of "f".
Exact sequences and regular functors.
In a regular category, a diagram of the form formula_1 is said to be an exact sequence if it is both a coequalizer and a kernel pair. The terminology is a generalization of exact sequences in homological algebra: in an abelian category, a diagram 
is exact in this sense if and only if formula_3 is a short exact sequence in the usual sense.
A functor between regular categories is called regular, if it preserves finite limits and coequalizers of kernel pairs. A functor is regular if and only if it preserves finite limits and exact sequences. For this reason, regular functors are sometimes called exact functors. Functors that preserve finite limits are often said to be left exact.
Regular logic and regular categories.
Regular logic is the fragment of first-order logic that can express statements of the form
formula_4,
where formula_5 and formula_6 are regular formulae i.e. formulae built up from atomic formulae, the truth constant, binary meets (conjunction) and existential quantification. Such formulae can be interpreted in a regular category, and the interpretation is a model of a sequent formula_4, if the interpretation of formula_8 factors through the interpretation of formula_9. This gives for each theory (set of sequents) "T" and for each regular category "C" a category Mod("T",C) of models of "T" in "C". This construction gives a functor Mod("T",-):RegCat→Cat from the category RegCat of small regular categories and regular functors to small categories. It is an important result that for each theory "T" there is a regular category "R(T)", such that for each regular category "C" there is an equivalence
formula_10,
which is natural in "C". Here, "R(T)" is called the "classifying" category of the regular theory "T." Up to equivalence any small regular category arises in this way as the classifying category of some regular theory.
Exact (effective) categories.
The theory of equivalence relations is a regular theory. An equivalence relation on an object formula_11 of a regular category is a monomorphism into formula_12 that satisfies the interpretations of the conditions for reflexivity, symmetry and transitivity. 
Every kernel pair formula_13 defines an equivalence relation formula_14. Conversely, an equivalence relation is said to be effective if it arises as a kernel pair. An equivalence relation is effective if and only if it has a coequalizer and it is the kernel pair of this.
A regular category is said to be exact, or exact in the sense of Barr, or effective regular, if every equivalence relation is effective. (Note that the term "exact category" is also used differently, for the exact categories in the sense of Quillen.)