Barrs Embedding Theorem for Enriched Categories

We generalize Barr’s embedding theorem for regular categories to the context of enriched categories.

💡 Research Summary

The paper “Barr’s Embedding Theorem for Enriched Categories” extends the classical embedding result of Barr, originally formulated for ordinary regular categories, to the setting of categories enriched over a monoidal base V. The authors begin by recalling Barr’s theorem: any small regular category C can be fully faithfully embedded into a large regular category of set‑valued presheaves, preserving finite limits and regular epimorphisms. This classical proof relies heavily on the existence of coequalizers, the stability of regular epimorphisms under pullback, and the factorisation of every morphism into a regular epi followed by a mono.

To transport these ideas into the enriched world, the paper introduces the notion of a V‑regular category. Here V is a complete, cocomplete, symmetric monoidal closed category (for instance, Ab, Vect, or a category of probability measures). A V‑category C is called V‑regular if it possesses V‑weighted colimits of a certain shape, V‑regular epimorphisms are stable under V‑pullbacks, and every V‑morphism admits a V‑image factorisation (a V‑regular epi followed by a V‑mono) that is unique up to V‑isomorphism. The authors show that these conditions are precisely what is needed to mimic the ordinary regularity arguments in the enriched context.

The central construction is the V‑enriched Yoneda embedding
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