Classifying strict discrete opfibrations with lax morphisms
Matteo Capucci, David Jaz Myers
TL;DR
The paper develops a general 2-categorical framework for classifying strict discrete opfibrations via lax morphisms in enhanced 2-categories. It introduces cartesianity for 2-algebras and shows how, under opfibrantly cartesian 2-monads and suitable exactness conditions, the classifying object $ ext{Omega}$ acquires a strict $T$-algebra structure that lifts to the enhanced 2-category $ extbf{Alg}_{ ext{lx}}(T)$, producing good 2-classifiers. A central lifting theorem connects the universality of discrete opfibration classifiers to their behavior under lax morphisms, and the work explains how to replace a monad by its pseudoalgebra coclassifier to achieve strictness in practical examples. The framework recovers and unifies known results for double categories, monoidal structures, and familial 2-monads, while providing new tools for iterated liftings and descent-like properties in preset presheaf contexts. Overall, the results yield concrete classifiers for strict opfibrations in a broad class of 2-algebraic theories, with significant implications for Yoneda-type theories in higher categorical settings.
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 Parè 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.