Gabriel-Ulmer duality is a contravariant biequivalence between small finite-limit categories and locally finitely presentable categories, which in the spirit of Lawvere's functorial semantics can be viewed as a *theory-model duality*: small finite-limit categories C are viewed as theories, and the lfp category FL(C, Set) of finite-limit preserving Set-valued functors is viewed as category of models of of C. The opposite of C can be reconstructed up to equivalence from Lex(C,Set) as full subcategory of *compact* objects.

The talk presents a *refinement* of Gabriel-Ulmer duality which is motivated by dependent type theory: finite-limit categories are replaced by *clans*, ie categories with a terminal object and a pullback-stable class of maps which is closed under composition, reflecting the structure of syntactic categories of *generalized algebraic theories* (GATs). The category of models of every GAT/clan is lfp, and to be able to reconstruct the clan we equip it with a *weak factorization system*, which in the classical case of ordinary algebraic theories (groups, rings, ...) consists of projective extensions on the left, and regular epimorphisms on the right.

Details can be found in the following preprint:

https://arxiv.org/abs/2308.11967

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