Arboreal categories and equi-resource homomorphism preservation theorems

The classical homomorphism preservation theorem, due to Los, Lyndon and Tarski, states that a first -order sentence phi is preserved under homomorphisms between structures if, and only if, it is equivalent to an existential positive sentence psi. Given a notion of (syntactic) complexity of sentences, an "equi-resource" homomorphism preservation theorem improves on the classical result by ensuring that psi can be chosen so that its complexity does not exceed that of phi. We describe an axiomatic approach to equi-resource homomorphism preservation theorems based on the notion of arboreal category. This framework is then employed to establish novel homomorphism preservation results, and improve on known ones, for various logic fragments, including first -order, guarded and modal logics. (c) 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).