Infinitely ludic categories

• Ano de defesa: 2024
• Autor(a) principal: Costa, Matheus Duzi Ferreira
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Categoria de jogos; Category theory; Fraïssé theory; Game category; Infinite game; Jogo infinito; Jogo topológico; Teoria das categorias; Teoria de Fraïssé; Topological game
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-29112024-201243/

This work is a presentation of a new approach to the study of infinite games in combinatorics. We introduce the categories GameA, GameB and, based on the existence of natural transformations, novel proofs to some classical results concerning topological games related to the duality between covering properties of X and convergence properties of Cp(X) are provided. We 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. These equivalent descriptions come with underlying functors to more familiar categories which help in the task of establishing some important properties of the game categories. Some of the presented categorical constructions have relevant game-theoretic interpretations. Following a recent trend in the field of combinatorics, we also explore a couple of perspectives on the Fraïssé theory of finite games, from which generalizations to the recently developed categorical Fraïssé theory arise.