The distributivity of limits over colimits is often a desirable trait for a category to possess. Included among the several examples of this behaviour, we find (infinitary) distributive categories, (infinitary) lextensive categories and (Grothendieck) toposes [2]. The theory of pseudomonads [1] is particularly well-suited to study the distributivity of limits over colimits of a given category. Indeed, this property is captured by the canonical pseudodistributive law between free limit completions and free colimit completions. Our goal is to give an overview of [4, 3], which studies the distributivity of limits over colimits with the tools of pseudodistributive laws. Many interesting classes of categories arise as the pseudoalgebras for the pseudomonads induced by these laws, and more were introduced by virtue of this approach. We also give some remarks on the emergence of exponentiable objects in the freely generated pseudoalgebras.

[1] F. Marmolejo. Distributive laws for pseudomonads. Theory Appl. Categ., 5(5):91-147, 1999.

[2] F. Marmolejo, R. Rosebrugh, R. Wood. Completely and totally distributive categories I. J. Pure Appl. Algebra, 216(8-9):1775-1790, 2012.

[3] F. Lucatelli Nunes, R. Prezado, M. Vákár. Free extensivity via distributivity. Port. Math., published online first. 2024.

[4] F. Lucatelli Nunes, M. Vákár. Free Doubly-Infinitary Distributive Categories are Cartesian Closed. DMUC Preprints, 24-16, 2024.