At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.
A category C is called regular if it satisfies the following three properties:[1]
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. In a regular category, a diagram of the form
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
is a short exact sequence in the usual sense.
A functor is regular if and only if it preserves finite limits and exact sequences.
Functors that preserve finite limits are often said to be left exact.
that satisfies the interpretations of the conditions for reflexivity, symmetry and transitivity.
Conversely, an equivalence relation is said to be effective if it arises as a kernel pair.
[3] 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.