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 mathe`matique (nouvelle s`

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

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 $E : C \longrightarrow [T, V]$.

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