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| In [[category theory]], a '''regular category''' is a category with [[limit (category theory)|finite limits]] and [[coequalizer]]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.
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| == Definition ==
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| A category ''C'' is called '''regular''' if it satisfies the following three properties:<ref>Pedicchio & Tholen (2004) p.177</ref>
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| * ''C'' is [[finitely complete category|finitely complete]].
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| * If ''f:X→Y'' is a [[morphism]] in ''C'', and
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| <div style="text-align: center;"> [[Image:Regular_category_1.png]] </div>
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| : is a [[pullback (category theory)|pullback]], then the coequalizer of ''p<sub>0</sub>,p<sub>1</sub>'' exists. The pair (''p<sub>0</sub>,p<sub>1</sub>'') is called the '''kernel pair''' of ''f''. Being a pullback, the kernel pair is unique up to a unique [[isomorphism]].
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| * If ''f:X→Y'' is a morphism in ''C'', and
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| <div style="text-align: center;"> [[Image:Regular_category_2.png]] </div>
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| : 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.
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| == Examples ==
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| Examples of regular categories include:
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| * '''[[Category of sets|Set]]''', the category of [[Set (mathematics)|sets]] and [[function (mathematics)|function]]s between the sets
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| * More generally, every elementary [[topos]]
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| * '''Grp''', the category of [[Group (mathematics)|groups]] and [[group homomorphism]]s
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| * The category of [[Field (mathematics)|fields]] and [[ring homomorphism]]s
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| * Every [[Semilattice|bounded join-semilattice]], with morphisms given by the order relation
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| * [[Abelian categories]]
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| The following categories are ''not'' regular:
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| * '''Top''', the category of [[topological space]]s and [[Continuous function (topology)|continuous function]]s
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| * '''Cat''', the category of [[small category|small categories]] and [[functor]]s
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| == Epi-mono factorization ==
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| In a regular category, the regular-[[epimorphism]]s and the [[monomorphism]]s 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''.
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| == Exact sequences and regular functors ==
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| In a regular category, a diagram of the form <math>R\rightrightarrows X\to Y</math> 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
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| :<math>R\overset r{\underset s\rightrightarrows} X\to Y</math>
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| is exact in this sense if and only if <math>0\to R\xrightarrow{r-s}X\to Y\to 0</math> is a [[short exact sequence]] in the usual sense.
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| 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'''.
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| == Regular logic and regular categories ==
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| Regular logic is the fragment of [[first-order logic]] that can express statements of the form
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| <center><math>\forall x (\phi (x) \to \psi (x))</math>,</center>
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| where <math>\phi</math> and <math>\psi</math> are regular [[Formula (mathematical logic)|formulae]] i.e. formulae built up from [[atomic formula]]e, the truth constant, binary [[Meet (mathematics)|meets]] and [[existential quantification]]. Such formulae can be interpreted in a regular category, and the interpretation is a model of a [[sequent]] | |
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| <center><math>\forall x (\phi (x) \to \psi (x))</math>,</center>
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| if the interpretation of <math>\phi </math> factors through the interpretation of <math> \psi</math>. 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 category|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
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| <center><math>\mathbf{Mod}(T,C)\cong \mathbf{RegCat}(R(T),C)</math>,</center>
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| which is natural in ''C''. Up to equivalence any small regular category ''C'' arises this way as the ''classifying'' category, of a regular theory.
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| == Exact (effective) categories ==
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| The theory of [[equivalence relations]] is a regular theory. An equivalence relation on an object <math>X</math> of a regular category is a monomorphism into <math>X \times X</math> that satisfies the interpretations of the conditions for reflexivity, symmetry and transitivity.
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| Every [[kernel pair]] <math>p_0, p_1: R \rightarrow X</math> defines an equivalence relation <math>R \rightarrow X \times X</math>. Conversely, an equivalence relation is said to be '''effective''' if it arises as a kernel pair.<ref>Pedicchio & Tholen (2004) p.169</ref> An equivalence relation is effective if and only if it has a coequalizer and it is the kernel pair of this.
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| 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.<ref>Pedicchio & Tholen (2004) p.179</ref>
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| ===Examples of exact categories ===
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| * 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]].
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| * Every [[abelian category]] is exact.
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| * Every category that is [[monad (category theory)|monadic]] over the category of sets is exact.
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| * The category of [[Stone space]]s is exact.
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| ==See also==
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| * [[Allegory (category theory)]]
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| * [[Topos]]
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| == References ==
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| {{reflist}}
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| * [[Michael_Barr_%28mathematician%29|Michael Barr]], Pierre A. Grillet, Donovan H. van Osdol. ''Exact Categories and Categories of Sheaves'', Springer, Lecture Notes in Mathematics 236. 1971.
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| * Francis Borceux, ''Handbook of Categorical Algebra 2'', Cambridge University Press, (1994).
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| * 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).
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| * Carsten Butz (1998), ''[http://www.brics.dk/LS/98/2/ Regular Categories and Regular Logic]'', BRICS Lectures Series LS-98-2, (1998).
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| * 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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| * {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}
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| [[Category:Category theory]]
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Andrew Simcox is the title his mothers and fathers gave him and he totally loves this title. To climb is something I really appreciate doing. Alaska is exactly where he's always been residing. She functions as a travel agent but quickly she'll be on her personal.
Also visit my blog; psychics online (visit the up coming document)