To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Regular category

From Wikipedia, the free encyclopedia

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.

YouTube Encyclopedic

  • 1/3
    3 068
  • WRO 2013-Regular Category (Junior High School)
  • WRO 2017 Regular Junior High Full Solution
  • WRO 2013 Regular Category - elementary level



A category C is called regular if it satisfies the following three properties:[1]

is a pullback, then the coequalizer of p0, p1 exists. The pair (p0p1) 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 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 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 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


where and 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 , if the interpretation of factors through the interpretation of .[2] 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,-):RegCatCat 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


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.[2]

Exact (effective) categories

The theory of equivalence relations is a regular theory. An equivalence relation on an object of a regular category is a monomorphism into that satisfies the interpretations of the conditions for reflexivity, symmetry and transitivity.

Every kernel pair defines an equivalence relation . 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.[4] (Note that the term "exact category" is also used differently, for the exact categories in the sense of Quillen.)

Examples of exact categories

See also


  1. ^ Pedicchio & Tholen 2004, p. 177
  2. ^ a b Butz, Carsten (1998). "Regular Categories and Regular Logic". BRICS Lectures Series LS-98-2.
  3. ^ Pedicchio & Tholen 2004, p. 169
  4. ^ Pedicchio & Tholen 2004, p. 179
This page was last edited on 29 October 2022, at 09:46
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.