Toward a characterization of algebraic exactness. (Q1427370)

Let \({\mathcal V}ar\) be the 2-category of many-sorted finitary algebraic categories called algebraic varieties, of algebraic functors and natural transformations. It is a 2-subcategory of the 2-category \(\mathcal{CAT}\) which generated a full 2-equation hull in \(\mathcal{CAT}\): the 2-category \(\mathcal{ALG}\) of so-called algebraically exact categories, algebraically exact functors, and natural transformations. It is proved that a functor between algebraically exact categories is algebraically exact if and only if it is continuous, finitary and exact. It is proved that a category with finite coproducts is algebraically exact if and only if: it is complete, has filtered colimits and reflexive coequalizers, is exact, in which products of regular epimorphisms are regular epimorphisms and filtered colimits commute with finite limits and distribute over products.