Objective 1 has been fully met, mainly through the work of Gehrke, Jakl and Reggio on a duality theoretic view on limits of finite structures. The main results were put in a broader context in a subsequent book chapter by the same authors, which offers a gentle introduction to this research area for a wider audience of mathematicians and computer scientists.
Objective 2 has been fully met. In particular, the work by Luca Reggio on polyadic sets and homomorphism counting, and on a duality perspective on the Beth definability property for the dual of compact Hausdorff spaces, provide new tools for the applicability of duality methods beyond finite structures.
Objective 3 has been fully met, especially through the work of Dawar, Jakl and Reggio on game comonads and Lovász-type results in finite model theory, and the work of Abramsky and Reggio on arboreal categories. The first makes the link between homomorphism counting results in finite model theory, game comonads, and polyadic sets—which are closely related to hyperdoctrines in categorical logic. The second provides an axiomatic approach to game comonads, yielding a unified treatment of several important aspects of resource-sensitive logics and paving the way for a systematised view of different categorical semantics for predicate logic.
Given the interdisciplinary nature of the project, an effort has been made to disseminate the results to as wide an audience as possible. Thus, the main results of this project have been presented at both mathematics and computer science conferences. These include two invited talks in international conferences and workshops (BLAST 2021 and the Resources and co-Resources workshop) and nine talks in international conferences (Logic Colloquium, ICALP, LiCS) and local seminars. Further, in August 2022 Jakl and Reggio will teach a course at the ESSLLI Summer School on game comonads, in which some of the results of this project will be presented.
