Regular category

In , a regular category is a category with 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. A category C is called regular if it satisfies the following three properties: C is . If f : X → Y is a morphism in C, and is a , then the coequalizer of p0, p1 exists. The pair (p0, p1) is called the kernel pair of f. Being a pullback, the kernel pair is unique up to a unique isomorphism. If f : X → Y is a morphism in C, and is a pullback, and if f is a regular epimorphism, then g is a regular epimorphism as well. A regular epimorphism is an epimorphism that appears as a coequalizer of some pair of morphisms. Examples of regular categories include: the category of sets and functions between the sets More generally, every elementary topos the category of groups and group homomorphisms The category of rings and ring homomorphisms More generally, the category of models of any variety Every bounded meet-semilattice, with morphisms given by the order relation Every abelian category The following categories are not regular: the category of topological spaces and continuous functions the category of and functors 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 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.