Regular category

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In 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. Regular categories were first introduced by Michael Barr in 1972.

1 Definition
2 Examples
3 Unique epi-mono factorization
4 Regular functors
5 Regular logic and regular categories
6 References

[edit] Definition

A category C is called regular if it satisfies the following three properties:

C has all finite limits.
If f:X→Y is a morphism in 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 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 which appears as a coequalizer of some pair of morphisms.

[edit] Examples

Examples of regular categories include:

The category Set of sets and functions between the sets.
The category Grp of groups and group homomorphisms.
The category of fields and ring homomorphism
Every poset with the order relation as morphisms.
Abelian categories

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 functors.

[edit] Unique epi-mono factorization

In a regular category, every morphism f:X→Y can be factorized into an 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 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.

[edit] Regular functors

A functor between regular categories is called regular, if it preserves finite limits and coqequalizers of kernel pairs.

[edit] Regular logic and regular categories

Regular logic is the fragment of first order logic that can express statements of the form

$\forall x (\phi (x) \to \psi (x))$ ,

where $φ$ and $ψ$ 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

$\forall x (\phi (x) \to \psi (x))$ ,

if the interpretation of $φ$ factors through the interpretation of $ψ$ . 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 category C arises this way as the classifying category, of a regular theory.

[edit] References

Francis Borceux, Handbook of Categorical Algebra 2, Cambridge University Press, (1994).
Stephen Lack, Theory and Applications of Categories, Vol.5, No.3, (1999).
Carsten Butz (1998), Regular Categories and Regular Logic, BRICS Lectures Series LS-98-2, (1998).
Jaap van Oosten (1995), Basic Category Theory, BRICS Lectures Series LS-95-1, (1995).

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Category: Category theory