Regular category
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Definition
A category C is called regular if it satisfies the following three properties:[1]
- C is finitely complete.
- 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 that appears as a coequalizer of some pair of morphisms.
Examples
Examples of regular categories include:
- Set, the category of sets and functions between the sets
- More generally, every elementary topos
- Grp, the category of groups and group homomorphisms
- The category of rings and ring homomorphisms
- More generally, the category of models of any variety
- Every bounded meet-semilattice, with morphisms given by the order relation
- Every abelian category
The following categories are not regular:
- continuous functions
- 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 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
is exact in this sense if and only if is a
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
where and are regular
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.[2]
Exact (effective) categories
The theory of
Every kernel pair defines an equivalence relation . Conversely, an equivalence relation is said to be effective if it arises as a kernel pair.[3] 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.[4] (Note that the term "exact category" is also used differently, for the exact categories in the sense of Quillen.)
Examples of exact categories
- The equivalence classes.
- Every abelian category is exact.
- Every category that is monadic over the category of sets is exact.
- The category of Stone spaces is regular, but not exact.
See also
- Allegory (category theory)
- Topos
- Exact completion
References
- ^ Pedicchio & Tholen 2004, p. 177
- ^ a b Butz, Carsten (1998). "Regular Categories and Regular Logic". BRICS Lectures Series LS-98-2.
- ^ Pedicchio & Tholen 2004, p. 169
- ^ Pedicchio & Tholen 2004, p. 179
- ISBN 978-3-540-36999-8.
- Borceux, Francis (1994). Handbook of Categorical Algebra. Vol. 2. Cambridge University Press. ISBN 0-521-44179-X.
- Lack, Stephen (1999). "A note on the exact completion of a regular category, and its infinitary generalizations". Theory and Applications of Categories. 5 (3): 70–80.
- van Oosten, Jaap (1995). "Basic Category Theory" (PDF). University of Aarhus. BRICS Lectures Series LS-95-1.
- Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge: Zbl 1034.18001.