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 $\mathcal{C}$ has certain good properties, then it admits an embedding with certain other good properties into another structure $\mathcal{D}$ which is somehow more explicit than $\mathcal{C}$.
For example,
by Tychonoff’s embedding theorem, if a $T_0$ space $X$ is completely regular, then there exists an embedding of $X$ into a product of metric spaces
by the Whitney embedding theorem, if an abstract $d$-dimensional real manifold $M$ is smooth, then there exists an embedding, which is an embedding of smooth manifolds, of $M$ into the explicit real manifold that is $\mathbb{R}^{2d}$
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 $V$. 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:
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 $E : C \longrightarrow [T, V]$.
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