Barr proved a theorem about embedding regular categories into categories of small presheaves, and also a strengthening for Barr exact categories.
Barr’s embedding theorem has the classical form of many embedding theorems in mathematics: if a structure has certain good properties, then it admits an embedding with certain other good properties into another structure which is somehow more explicit than .
For example,
by Tychonoff’s embedding theorem, if a space is completely regular, then there exists an embedding of into a product of metric spaces
by the Whitney embedding theorem, if an abstract -dimensional real manifold is smooth, then there exists an embedding, which is an embedding of smooth manifolds, of into the explicit real manifold that is
by the Freyd-Mitchell embedding theorem, if a category is small and abelian, then there exist an embedding, which is exact, into the category of modules of a (not necessarily commutative) ring.
and,
The proof of (a version of) Barr’s theorem given by Makkai in Makkai1980 is a nice example of a non-trivial application of ultraproducts in category theory.
It has been proved in
and, in a different way, in
M. Barr, Representation of categories, J. Pure Appl. Alg. 41 (1986) 113-137 (this article has supposedly some fixable errors).
F. Borceux, A propos d’un théorème de Barr, Séminaire de mathématique (nouvelle série) Rapport 28 - Mai 1983, Institute de Mathematique Pure et Appliqee, Univ. Cath. de Louvain.
M. Makkai, A theorem on Barr-exact categories, with an infinitary generalization, Ann. Pure Appl. Logic 47 (1990), no. 3, 225-268.
Michael Barr‘s full exact embedding theorem for Barr exact categories, proved in (?)
generalizes the Lubkin-Freyd-Mitchell embedding theorems for abelian categories. The Giraud’s theorem for topoi is not much more than a special case of that theorem.
M. Makkai, On full embeddings I, Journal of Pure and Applied Algebra 16, (1980), pp. 183-195
M. Makkai, Full continuous embeddings of toposes, Trans. Amer. Math. Soc. 269, No. 1 (Jan., 1982), pp. 167-196 jstor
One can also consider categories enriched over a locally finitely presentable closed symmetric monoidal category . Such a -category is regular if it is finitely complete, admits the coequalizers of kernel pairs all regular epimorphisms are universal (i.e. stable under pullbacks) and stable under cotensors with the finite objects. A -functor is regular if it preserves finite limits and regular epimorphisms. The following generalizes the Barr’s embedding theorem for regular categories to the regular enriched categories:
Its main result is
Theorem 10. For a small regular -category there exists a small category and a regular fully faithful functor .
Last revised on July 20, 2017 at 13:01:27. See the history of this page for a list of all contributions to it.