- Regular category
category theory, a regular category is a category with finite limits and coequalizers of 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.
A category "C" is called regular if it satisfies the following three properties:
* "C" is finitely complete.
* If "f:X→Y" is a
morphismin "C", and
: is a pullback, then the coequalizer of "p0,p1" exists. The pair ("p0,p1") is called the kernel pair of "f". Being a pullback, the kernel pair is unique up to a unique
* 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 of regular categories include:
* The category Set of sets and functions between the sets.
* The category Grp of groups and
* The category of fields and
posetwith the order relation as morphisms.
The following categories are "not" regular:
* The category Top of
topological spaces and continuous functions is an example of a category which is not regular.
* The category Cat of small categories and
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 sequencesin homological algebra: in an abelian category, a diagram : is exact in this sense if and only if is a short exact sequencein 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 logicthat can express statements of the form
where and are regular formulae i.e. formulae built up from
atomic formulae, the truth constant, binary meets 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 . 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
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 relationsis 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.
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
category of setsis exact in this sense, and so is any (elementary) topos. Every equivalence relation has a coequalizer, which is found by taking equivalence classes.
abelian categoryis exact.
* Every category that is monadic over the category of sets is exact.
* The category of
Stone spaces is exact.
Allegory (category theory)
* 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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