Categorical Finite Model Theory: Game Comonads and Arboreal Categories
(Habilitation thesis, 2017/12/28)Abstract
This habilitation thesis surveys almost a decade of my research in the area of game comonads, which is a new research area in the intersection of categorical logic and finite model theory. The unifying idea running through this approach is that by formalising basic concepts of finite model theory in the language of comonads and elementary path categories (e.g. arboreal categories), we obtain an elegant abstract theory which (1) specialises to old and new theorems and examples and (2) reveals connections with other branches of category theory. We discuss applications of this approach to the study of homomorphism counting theorems, graph parameters, composition methods (e.g. the theorems of Mostowski and Feferman–Vaught) and model-theoretic types.
The thesis also contains an extensive introduction to the theory of game comonads, built step-by-step from relevant concepts of finite model theory and comonad theory. This introduction should be accessible to anyone with minimal knowledge of category theory and logic.
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BibTex
@unpublished{jakl2026,
title = {Categorical Finite Model Theory: Game Comonads and Arboreal Categories},
author = {Tom{\'a}{\v s} Jakl},
note = {Habilitation thesis},
year = {2026}
}