Duality for Clans: an Extension of Gabriel-Ulmer Duality
Jonas Frey
TL;DR
The paper addresses extending Gabriel–Ulmer duality to capture additional syntactic information in generalized algebraic theories by introducing clans, a categorical representation of GATs with a specified class of display maps. It builds a contravariant 2-functor from clans to structured categories endowed with a well-behaved weak factorization system, and proves that this biadjunction is idempotent. The central result is a biequivalence $Clan_{cc}^{op} \simeq ClanAlg$, thereby generalizing Adámek–Rosický–Vitale's duality and recovering it in a suitable specialization. As applications, slice and coslice categories of models are described as derived clan models, and further clan-representations for the category ${Cat}$ are obtained, illustrating the semantic and structural gains of the approach.
Abstract
Clans are representations of generalized algebraic theories that contain more information than the finite-limit categories associated to the locally finitely presentable categories of models via Gabriel-Ulmer duality. Extending Gabriel-Ulmer duality to account for this additional information, we present a duality theory between clans and locally finitely presentable categories equipped with a weak factorization system of a certain kind.