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Abstract
Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosický, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Working over a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of PrestRajani/KuberRosický.
Original language  English 

Article number  106268 
Pages (fromto)  129 
Number of pages  29 
Journal  Journal of Pure and Applied Algebra 
Volume  224 
Issue number  6 
DOIs  
Publication status  Published  Jun 2020 
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 1 Active

Working synthetically in higher categorical structures
Lack, S., Verity, D., Garner, R. & Street, R.
19/06/19 → 18/06/22
Project: Other