Infinitely ludic categories
Matheus Duzi, Paul Szeptycki, Walter Tholen
TL;DR
The paper builds a comprehensive categorical framework for infinite, perfect-information games by introducing two equivalent categories, $\mathbf{Game}_{A}$ and $\mathbf{Game}_{B}$, whose objects are infinite games and whose morphisms capture natural, chronology-respecting transformations. It uncovers multiple equivalent descriptions of these ludic categories (arboreal, presheaf-topos, and metric presentations) and proves a universality result: every game embeds into a Banach-Mazur game over a complete ultrametric space, revealing deep connections between topology and game dynamics. The work provides a thorough categorical analysis, establishing completeness, cocompleteness, and cartesian closedness, describing four orthogonal factorization systems, and showing both coregularity and the lack of local cartesian closedness; it also develops weak classifiers for strong partial maps and discusses metric interpretations of game constructions. By linking topological covering and convergence properties through natural transformations, the paper illuminates how familiar results (e.g., Scheepers dualities) admit a categorified proof within this framework. Overall, the results bridge infinite game theory, topos-theoretic methods, and metric geometry, offering a robust toolkit for analyzing and interpreting categorical structures in game-theoretic contexts.
Abstract
Pursuing a new approach to the study of infinite games in combinatorics, we introduce the categories $\mathbf{Game}_{A}$ and $\mathbf{Game}_{B}$ and improve some classical results concerning topological games related to the duality between covering properties of $X$ and convergence properties of $\mathrm{C}_{\mathrm {p}}(X)$ by establishing the existence and key role of certain natural transformations. We then describe these ludic categories in various equivalent forms, viewing their objects as certain structured trees, presheaves, or metric spaces, and we thereby obtain their arboreal, functorial and metrical appearances. We use their metrical disguise to demonstrate a universality property of the Banach-Mazur game. The various equivalent descriptions come with underlying functors to more familiar categories which help establishing some important properties of the game categories: they are complete, cocomplete, extensive, cartesian closed, and coregular, but neither regular nor locally cartesian closed. We prove that their classes of strong epimorphisms, of regular epimorphisms, and of descent morphisms, are all distinct, and we show that these categories have weak classifiers for strong partial maps. Some of the categorical constructions have interesting game-theoretic interpretations.