We prove a generalization of a theorem of Bunge and Gray about forming colax adjunctions out of relative Kan extensions and apply it to the study of the Kleisli 2-category for a lax-idempotent pseudomonad. For instance, we establish the weak completeness of the Kleisli 2-category and describe colax change-of-base adjunctions between Kleisli 2-categories. Our approach covers such examples as the bicategory of small profunctors and the 2-category of lax triangles in a 2-category. The duals of our results provide lax analogues of classical results in two-dimensional monad theory: for instance, establishing the weak cocompleteness of the 2-category of strict algebras and lax morphisms and the existence of colax change-of-base adjunctions.
Keywords: 2-category, lax adjunction, lax-idempotent pseudomonad, KZ-pseudomonad
2020 MSC: 18N10, 18N15, 18D60, 18D65, 18C20
Theory and Applications of Categories, Vol. 44, 2025, No. 7, pp 227-271.
Published 2025-02-06.
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