Accessibility and presentability in 2-categories

TL;DR

This paper develops a framework for understanding accessible and presentable objects in 2-categories with specific monadic structures, unifying various Gabriel-Ulmer theorems and illustrating with classical and advanced examples.

Contribution

It introduces a new definition of accessible and presentable objects in 2-categories with KZ contexts, and proposes the concept of Gabriel-Ulmer envelopes to establish dualities.

Findings

Unified treatment of Gabriel-Ulmer theorems in 2-categories

Introduction of Gabriel-Ulmer envelopes for KZ contexts

Examples spanning classical, enriched, and higher category theory

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.