- Regular category
In

category theory , a**regular category**is a category with finite limits andcoequalizer s of kernel pairs, satisfying certain "exactness" conditions. In that way, regular categories recapture many properties ofabelian categories , like the existence of "images", without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment offirst-order logic , known as regular logic.**Definition**A category "C" is called

**regular**if it satisfies the following three properties:

* "C" is finitely complete.

* If "f:X→Y" is amorphism in "C", and: 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 uniqueisomorphism .

* 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 which appears as a coequalizer of some pair of morphisms.**Examples**Examples of regular categories include:

* The category**Set**of sets and functions between the sets.

* The category**Grp**of groups andgroup homomorphism s.

* The category of fields andring homomorphism

* Everyposet with the order relation as morphisms.

*Abelian categories The following categories are "not" regular:

* The category**Top**oftopological space s and continuous functions is an example of a category which is not regular.

* The category**Cat**of small categories andfunctor s.**Epi-mono factorization**In a regular category, the regular-

epimorphism s and themonomorphism s form afactorization system . Every morphism "f:X→Y" can be factorized into a regularepimorphism "e:X→E" followed by amonomorphism "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 $R\; ightrightarrows\; X\; o\; Y$ is said to be an

**exact sequence**if it is both a coequalizer and a kernel pair. The terminology is a generalization ofexact sequences inhomological algebra : in anabelian category , a diagram :$Roverset\; r\{underset\; s\; ightrightarrows\}\; X\; o\; Y$ is exact in this sense if and only if $0\; o\; Rxrightarrow\{r-s\}X\; o\; Y\; o\; 0$ is ashort 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

$forall\; x\; (phi\; (x)\; o\; psi\; (x))$,where $phi$ and $psi$ are regular formulae i.e. formulae built up from

atomic formula e, the truth constant, binary meets andexistential quantification . Such formulae can be interpreted in a regular category, and the interpretation is a model of asequent

$forall\; x\; (phi\; (x)\; o\; psi\; (x))$,if the interpretation of $phi$ factors through the interpretation of $psi$. This gives for each theory (set of sequences) 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" and for each category "C", there is a category "R(T)" and an equivalence

$mathbf\{Mod\}(T,C)cong\; mathbf\{RegCat\}(R(T),C)$,which is natural in "C". Up to equivalence any small regular category "C" arises this way as the "classifying" category, of a regular theory.

**Exact (effective) categories**The theory of

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

kernel pair "p"_{0},"p"_{1}:"R"→"X" defines an equivalence relation "R"→"X"x"X". 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.**Examples of exact categories*** The

category of sets is exact in this sense, and so is any (elementary)topos . Every equivalence relation has a coequalizer, which is found by takingequivalence classes .* Every

abelian category is exact.* Every category that is monadic over the category of sets is exact.

* The category of

Stone space s is exact.**ee also***

Allegory (category theory)

*Topos **References*** Michael Barr, Pierre A. Grillet, Donovan H. van Osdol. "Exact Categories and Categories of Sheaves", Springer, Lecture Notes in Mathematics 236. 1971.

* Francis Borceux, "Handbook of Categorical Algebra 2", Cambridge University Press, (1994).

* Stephen Lack, " [*http://www.tac.mta.ca/tac/index.html#vol5 A note on the exact completion of a regular category, and its infinitary generalizations*] ". Theory and Applications of Categories, Vol.5, No.3, (1999).

* Carsten Butz (1998), " [*http://www.brics.dk/LS/98/2/ Regular Categories and Regular Logic*] ", BRICS Lectures Series LS-98-2, (1998).

* Jaap van Oosten (1995), " [*http://www.brics.dk/LS/95/1/BRICS-LS-95-1/BRICS-LS-95-1.html Basic Category Theory*] ", BRICS Lectures Series LS-95-1, (1995).

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