On algebraically exact categories and essential localizations of varieties (Q5952407)

The paper establishes important results, to be considered on the one hand as extensions of classical theorems on additive categories and essential localizations of module categories, and on the other hand as new insights into the modern world of algebraically exact categories, introduced by \textit{J. Adámek} and \textit{J. Rosický} as the categories with limits and sifted colimits which must resonably commute [Theory Appl. Categ. 8, 33-53 (2001; Zbl 0971.18004)]. Hence, among other things, the authors characterize the essential localizations of locally finitely-presentable categories as the complete and cocomplete categories with a regular generator in which finite limits commute with filtered colimits and products distribute over them. If one adds to these the condition of Barr-exactness and the preservation of regular epimorphisms by products, then one obtains precisely the essential localizations of varieties of finitary general algebras.