Categories of Games and their Fraïssé Theory
Matheus Duzi, Paul Szeptycki, Walter Tholen
TL;DR
The paper addresses how Fraïssé theory can be extended to categories of abstract finite two-player games by adopting a categorical framework. It first shows the classical model-theoretic approach does not apply to these games and then constructs a Fraïssé limit $G_{ ext{Flim}}$ as the direct limit of a Fraïssé sequence in the category of finite games with embeddings, proving $G_{ ext{Flim}}$ is countable and ultrahomogeneous with respect to embeddings into finite games. A key contribution is the characterization of weakly finitely small games (finite and Alice-trivial) and a general categorical path to ultrahomogeneity via tight squeeze properties, enabling a broader Fraïssé theory for games. A KPT-style analysis then relates the automorphism group of $G_{ ext{Flim}}$ to topological dynamics, showing non-extreme amenability and highlighting open questions about Ramsey properties for finite games and their dynamical consequences.
Abstract
Relying on recent generalizations of the Fraïssé theory to a broader category-theoretic context, we study the class of abstract finite games played between two players and show the existence of an infinitetly countable game which is ultrahomogeneous and universal with respect to said class. Certain peculiarities of our game categories which clash with the usual framework found in the literature then lead us to formulate weaker category-theoretic properties which still yield a universal and ultrahomogeneous Fraïssé limit, thus further generalizing the categorical framework for a Fraïssé theory.