Finitely accessible arboreal adjunctions and Hintikka formulae

Abstract

Arboreal categories provide an axiomatic framework in which abstract notions of bisimilarity and back-and-forth games can be defined. They act on extensional categories, typically consisting of relational structures, via arboreal adjunctions. In many cases, equivalence of structures in fragments of infinitary first-order logic can be captured by transferring the bisimilarity relation along the adjunction. In most applications, the categories involved are locally finitely presentable and the adjunctions are finitely accessible. Our main result identifies the expressive power of this class of adjunctions. We show that the ranks of back-and-forth games in the arboreal category are definable by formulae \`a la Hintikka, and thus the relation between extensional objects induced by bisimilarity is always coarser than equivalence in infinitary first-order logic. Our approach leverages Gabriel-Ulmer duality for locally finitely presentable categories, and Hodges' word-constructions.