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 $V$. Such a $V$-category $C$ 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 $V$-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 $V$-category $C$ there exists a small category $T$ and a regular fully faithful functor $E:C⟶[T,V]$.