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
and, in a different way, in
matique (nouvelle srie) Rapport 28 - Mai 1983, Institute de Mathematique Pure et Appliqee, Univ. Cath. de Louvain.
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.
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 .