Categories of Games and their Fra\"iss\'e Theory

TL;DR

This paper extends Fra"iss"e theory to a broader categorical context, constructing a universal, ultrahomogeneous infinite game that generalizes classical results for finite games.

Contribution

It introduces a generalized categorical Fra"iss"e framework and constructs a universal, ultrahomogeneous infinite game within this setting.

Findings

Existence of a universal, ultrahomogeneous infinite game.

Generalization of Fra"iss"e theory to broader category-theoretic contexts.

Identification of weaker categorical properties still yielding Fra"iss"e limits.

Abstract

Relying on recent generalizations of the Fra\"iss\'e 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 Fra\"iss\'e limit, thus further generalizing the categorical framework for a Fra\"iss\'e theory.