Dualities in the theory of accessible categories
Giacomo Tendas
TL;DR
This work unifies and extends dualities for accessible ${\mathcal{V}}$-categories by introducing weakly sound classes of weights ${\Psi}$ and proving a biequivalence between the 2-category of $\alpha$-accessible ${\mathcal{V}}$-categories with ${\Psi}$-limits and ${\Psi^+}$-free ${\mathcal{V}}$-categories free under ${\Psi^+}$-colimits. This framework recovers classical dualities such as Gabriel–Ulmer, Diers, and Makkai–Paré in enriched form, while also enabling new dualities (e.g., bi-involution for $\alpha$-accessible ${\mathcal{V}}$-categories). The paper further develops an enriched Scott adjunction between accessible ${\mathcal{V}}$-categories with flat/filtered colimits and ${\mathcal{V}}$-topoi, showing how ${\mathcal{V}}$-topoi arise as left exact localizations of presheaf ${\mathcal{V}}$-categories. Overall, the work provides a formal, unifying enrichment-based approach to dualities in category theory and suggests directions for broader generalizations via the theory of companions.
Abstract
Through the notion of weakly sound class of weights, we recover many known dualities involving accessible categories with a chosen class of limits, as instances of a general duality theorem. These include the Gabriel-Ulmer duality for locally finitely presentable categories, Diers duality for locally finitely multipresentable categories, and the Makkai-Paré duality for finitely accessible categories. In doing so, we extend these to the enriched setting, provide a more formal and unifying approach to the theory, and also discuss new dualities that arise as a consequence of our main theorem.