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In
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
, a regular category is a category with finite limits and
coequalizer In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is a ...
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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quant ...
, 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 a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphism ...
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 unique
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. * If ''f'' : ''X'' → ''Y'' is a morphism in ''C'', and
: is a pullback, and if ''f'' is a regular
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
, 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: * Set, the category of sets and
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orien ...
s between the sets * More generally, every elementary
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
* Grp, the category of groups and
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
s * The category of
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film an ...
and
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preserv ...
s * More generally, the category of models of any
variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film) ...
* Every bounded meet-semilattice, with morphisms given by the order relation * Every
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abel ...
The following categories are ''not'' regular: *
Top A spinning top, or simply a top, is a toy with a squat body and a sharp point at the bottom, designed to be spun on its vertical axis, balancing on the tip due to the gyroscopic effect. Once set in motion, a top will usually wobble for a few ...
, the category of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
s and
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s *
Cat The cat (''Felis catus'') is a domestic species of small carnivorous mammal. It is the only domesticated species in the family Felidae and is commonly referred to as the domestic cat or house cat to distinguish it from the wild members of ...
, the category of small categories and
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
s


Epi-mono factorization

In a regular category, the regular-
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
s and the
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphi ...
s form a factorization system. Every morphism ''f:X→Y'' can be factorized into a regular
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
''e:X→E'' followed by a
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphi ...
''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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
''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 An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the contex ...
in
homological algebra Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topol ...
: in an
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abel ...
, 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 An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the contex ...
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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quant ...
that can express statements of the form where \phi and \psi are regular
formulae In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a '' chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship bet ...
i.e. formulae built up from
atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformu ...
e, the truth constant, binary meets (conjunction) and
existential quantification In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, whe ...
. Such formulae can be interpreted in a regular category, and the interpretation is a model of a
sequent In mathematical logic, a sequent is a very general kind of conditional assertion. : A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n. A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of ass ...
\forall x (\phi (x) \to \psi (x)), if the interpretation of \phi factors through the interpretation of \psi. 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 Small may refer to: Science and technology * SMALL, an ALGOL-like programming language * Small (anatomy), the lumbar region of the back * ''Small'' (journal), a nano-science publication * <small>, an HTML element that defines smaller text ...
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.


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
Barr Barr may refer to: Places * Barr (placename element), element of place names meaning 'wooded hill', 'natural barrier' * Barr, Ayrshire, a village in Scotland * Barr Building (Washington, DC), listed on the US National Register of Historic Place ...
, or effective regular, if every equivalence relation is effective. (Note that the term "exact category" is also used differently, for the exact categories in the sense of Quillen.)


Examples of exact categories

* The
category of sets In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of ...
is exact in this sense, and so is any (elementary)
topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
. Every equivalence relation has a coequalizer, which is found by taking equivalence classes. * Every
abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abel ...
is exact. * Every category that is monadic over the category of sets is exact. * The category of
Stone space In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in ...
s is regular, but not exact.


See also

* Allegory (category theory) *
Topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion ...
* Exact completion


References

* * * * * {{refend Categories in category theory