Accessibility and presentability in 2-categories

Abstract

We outline a definition of accessible and presentable objects in a 2-category $\mathcal{K}$ endowed with a "KZ context", that is to say a pair of lax-idempotent monads interacting in a prescribed way; this perspective suggests a unified treatment of many "Gabriel-Ulmer like" theorems, asserting how presentable objects arise as reflections of generating ones. We outline the notion of "(Gabriel-Ulmer) envelope" for a KZ context, sufficient to concoct Gabriel-Ulmer duality. We end the paper with a roundup of examples, involving classical (set-based and enriched), low dimensional category theory, and a perspective for future work, rooted in higher category theory and homotopy theory.

Final version, accepted for publication on JPAA

References in corpus (6)

• Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures
• Abstract tilting theory for quivers and related categories
• Yoneda structures and KZ doctrines
• Fibrations and Yoneda's lemma in an $\infty$-cosmos
• Localization theory for derivators
• On the unicity of formal category theories

Cited by in corpus (6)

• Accessible categories with a class of limits
• On the unicity of formal category theories
• Adjoint functor theorems for lax-idempotent pseudomonads
• Dualities in the theory of accessible categories
• Gabriel-Ulmer duality for topoi and its relation with site presentations
• More on soundness in the enriched context