Classifying strict discrete opfibrations with lax morphisms

Abstract: We study how discrete opfibration classifiers in a(n enhanced) 2-category can be endowed with the structure of a $T$-algebra and thereby lift to the enhanced 2-category of 2-algebras and lax morphisms. To support this study, we give a definition of discrete opfibration classifier in the enhanced setting in which tight (e.g. strict) discrete opfibrations are classified by loose (e.g. lax) maps. We then single out conditions on the 2-monad $T$ and the classifier that make this possible, and observe these hold in a wide range of examples: double categories (recovering the results of Par`e and Lambert), (symmetric) monoidal categories, and all structures encoded by familial 2-monads. We also prove the properties needed on such 2-monads are stable under replacement by pseudo-algebra coclassifiers (when sufficient exactness conditions hold), allowing us to replace a pseudo-algebra structure on the classifier by a strict one. To get to our main theorem, we introduce the concepts of \emph{cartesian maps} and \emph{cartesian objects} of a 2-algebra, which generalize various other notions in category theory such as cartesian monoidal categories, extensive categories, categories with descent, and more. As a corollary, we characterize when representable copresheaves are pseudo rather than lax in terms of the cartesianity at their representing object.