Localizations of algebraic categories. II (Q1295671)

The author had previously characterized the localizations of monadic categories over \({\mathcal S}{\mathcal E}{\mathcal T}\) as the Barr-exact categories with a regular generator and all of its copowers [see part I, \textit{E. M. Vitale}, J. Pure Appl. Algebra 108, No. 3, 315-320 (1996; Zbl 0854.18007)]. By adding to these conditions the requirement that directed colimits commute with finite limits, he has now obtained a characterization of localizations of algebraic categories in the sense of Lawvere. This result also leads to a new proof of the Popescu-Gabriel characterization of Grothendieck categories: the abelian categories with a generator and its copowers in which directed colimits exist and commute with finite limits are precisely the localizations of module categories.