regular category

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In
category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
, a regular category is a category with finite limits and
coequalizer In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled ...
s 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.

# Definition

A category ''C'' is called regular if it satisfies the following three properties: * ''C'' is finitely complete category, finitely complete. * If ''f'' : ''X'' → ''Y'' is a morphism in ''C'', and
: is a pullback (category theory), 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. A regular epimorphism is an epimorphism that appears as a coequalizer of some pair of morphisms.

# Examples

Examples of regular categories include: * Category of sets, Set, the category of Set (mathematics), sets and function (mathematics), functions between the sets * More generally, every elementary topos * Category of groups, Grp, the category of Group (mathematics), groups and group homomorphisms * The category of Ring (mathematics), rings and ring homomorphisms * More generally, the category of models of any Variety (universal algebra), variety * Every Semilattice, bounded meet-semilattice, with morphisms given by the order relation * Every Abelian categories, abelian category The following categories are ''not'' regular: * Category of topological spaces, Top, the category of topological spaces and Continuous function (topology), continuous functions * Category of small categories, Cat, the category of small category, small categories and functors

# 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 category (mathematics)#Types of morphisms, 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 $R\rightrightarrows X\to Y$ 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 :$R\;\overset r\; X\xrightarrow Y$ is exact in this sense if and only if $0\to R\xrightarrowX\oplus X\xrightarrow Y\to 0$ 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
$\forall x \left(\phi \left(x\right) \to \psi \left(x\right)\right)$,
where $\phi$ and $\psi$ are regular Formula (mathematical logic), formulae i.e. formulae built up from atomic formulae, the truth constant, binary Meet (mathematics), meets (conjunction) and existential quantification. Such formulae can be interpreted in a regular category, and the interpretation is a model of a sequent $\forall x \left(\phi \left(x\right) \to \psi \left(x\right)\right)$, if the interpretation of $\phi$ factors through the interpretation of $\psi$.Carsten Butz (1998),
Regular Categories and Regular Logic
', BRICS Lectures Series LS-98-2, (1998).
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'',-):RegCat→Cat from the category RegCat of small category, 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 of categories, equivalence
$\mathbf\left(T,C\right)\cong \mathbf\left(R\left(T\right),C\right)$,
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.

# 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 $X \times X$ that satisfies the interpretations of the conditions for reflexivity, symmetry and transitivity. Every kernel pair $p_0, p_1: R \rightarrow X$ defines an equivalence relation $R \rightarrow X \times X$. 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 Michael Barr (mathematician), Barr, or effective regular, if every equivalence relation is effective.Pedicchio & Tholen (2004) p.179 (Note that the term "exact category" is also used differently, for the Exact category, exact categories in the sense of Quillen.)

## 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 taking equivalence classes. * Every abelian category is exact. * Every category that is monad (category theory), monadic over the category of sets is exact. * The category of Stone spaces is regular, but not exact.