A simple characterization of theories of varieties
It is known [\textit{J. Adámek} and \textit{H.-E. Porst}, J. Algebra 208, No. 2, 379-398 (1998; Zbl 0913.18004)] that the theories whose categories of models are quasivarieties are precisely the finitely cocomplete categories with enough regular projectives. It is also known [\textit{A. Carboni, M. C. Peddichio}, and \textit{J. Rosický}, J. Pure Appl. Algebra 161, No. 1-2, 65-90 (2001; Zbl 0982.18006)] that to exact categories regarded as categories of models correspond certain types of theories called ``proexact'' but which are rather technically defined. Since the varieties are the exact quasivarieties, a combination of the above two results readily gives a characterization of the theories of varieties. What the authors show in this paper is that a simpler approach is more fruitful and prove that the theories whose categories of models are the varieties are precisely the finitely cocomplete categories having enough effective projectives, by which they mean objects \(E\) such that \(\hom(E,-)\) preserves coequalizers of reflexive graphs. They also exhibit the 2-category of varieties as a bireflective sub 2-category of the 2-category of locally finitely presentable categories.
scientific article; zbMATH DE number 1556966