On Algebraically Exact Categories and Essential Localizations of Varieties

Algebraically exact categories have been introduced in J. Adámek, F. W. Lawvere, and J. Rosický (to appear), as an equational hull of the 2-category VAR of all varieties of finitary algebras. We will show that algebraically exact categories with a regular generator are precisely the essential localizations of varieties and that, in this case, algebraic exactness is equivalent to (1) exactness, (2) commutativity of filtered colimits with finite limits, (3) distributivity of filtered colimits over arbitrary products, and (4) product-stability of regular epimorphisms. This can be viewed as a nonadditive generalization of the classical Roos Theorem characterizing essential localizations of categories of modules. Analogously, precontinuous categories, introduced in J. Adámek, F. W. Lawrence, and J. Rosický (to appear) as an equational hull of the 2-category LFP (of locally finitely presentable categories), are characterized by the above properties (2) and (3). Essential localizations of locally finitely presentable categories and presheaf categories are fully described.