nLab
Barr embedding theorem

Barr proved an embedding theorem for regular categories and a strengthening for the Barr exact categories.

Barr’s embedding theorem for regular categories says that every small regular category can be embedded into a category of small presheaves. It has been proved

  • M. Barr, Exact categories, Lecture Notes in Math. 236, (Springer, Berlin, 1971), 1-119.

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 theoreme de Barr, Se’minaire de mathematique (nouvelle srie) 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

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, 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 September 24, 2012 18:32:12 by Tim Porter (95.147.237.35)