Regular category

In category theory, a regular category is a category with finite limits and coequalizers of all pairs 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.

In category theory, a regular category is a category with finite limits and coequalizers of all pairs 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. is a pullback, then the coequalizer of p 0 , p 1 exists. The pair (p 0 , p 1 ) 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. 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. ==See also== Allegory (category theory) Coherent category Topos Exact completion

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