How algebraic is algebra? (Q2710150)

The inclusion functor of the 2-category \({\mathcal V}{\mathcal A}{\mathcal R}\) of all finitary varieties (with finitary right adjoint functors preserving regular epimorphisms as arrows, and with natural transformations as 2-cells) into the (larger) 2-category \({\mathcal C}{\mathcal A}{\mathcal T}\) of all categories fails to create split coequalizers and therefore fails to be (pseudo)monadic. Hence, the problem at hand is to determine the (reasonably defined) equational hull of \({\mathcal V}{\mathcal A}{\mathcal R}\) in \({\mathcal C}{\mathcal A}{\mathcal T}\).NEWLINENEWLINENEWLINEThe authors offer a solution under the restriction to ranked operations: it is the 2-category \({\mathcal A}{\mathcal L}{\mathcal G}\) of algebraically exact categories, i.e., of complete categories with sifted colimits which distribute over limits, with functors preserving limits and sifted colimits as arrows, and with all natural transformations. \({\mathcal A}{\mathcal L}{\mathcal G}\) is bi-equivalent to the 2-category of algebras of the composite pseudomonad \(Sind\circ Lim\), where \textit{Lim} is the free-completion 2-monad under limits and \textit{Sind} that one under sifted colimits; the pseudomonad exists since \textit{Sind} (Beck-)distributes over \textit{Lim}. The authors also mention a number of challenging open problems.