Barr embedding theorem


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 𝒞\mathcal{C} has certain good properties, then it admits an embedding with certain other good properties into another structure 𝒟\mathcal{D} which is somehow more explicit than 𝒞\mathcal{C}.

For example,


A proof of Makkai

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

  • M. Barr, Exact categories, Lecture Notes in Math. 236, (Springer, Berlin, 1971), 1-120.

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 thĂ©orème de Barr, SĂ©minaire de mathĂ©matique (nouvelle sĂ©rie) 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 (?)

  • Michael Barr, Embedding of categories, Proc. Amer. Math. Soc. 37, No. 1 (Jan., 1973), pp. 42-46 (jstor, pdf)

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 VV. Such a VV-category CC 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 VV-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:

  • Dimitri Chikhladze, Barr’s embedding theorem for enriched categories, J. Pure Appl. Alg. 215, n. 9 (2011) 2148-2153, arxiv/0903.1173, doi

Its main result is

Theorem 10. For a small regular VV-category CC there exists a small category TT and a regular fully faithful functor E:C⟶[T,V]E : C \longrightarrow [T, V].

Revised on July 20, 2017 09:01:27 by Urs Schreiber (