The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we give a proof for the effective Banach Category Theorem. Further, we provide a game-theoretic proof of an effective theorem in dynamical systems, namely the category version of Poincaré Recurrence. The Poincaré Recurrence Theorem for category states that for a homeomorphism without open wandering sets, the set of non recurrent points forms a first category (meager) set. As an application of the effectivization of the Banach-Mazur game, we show that such a result holds true in effective settings as well.

翻译：经典的Banach-Mazur博弈刻画了拓扑空间中的第一范畴集。本文证明该博弈的有效化版本可刻画有效第一范畴集，并据此给出有效Banach范畴定理的证明。进一步，我们为动力系统中的一个有效定理——即范畴论版本的Poincaré回归定理——提供博弈论证明。范畴论Poincaré回归定理指出：对于无开游荡集的同胚映射，非回归点集构成第一范畴（贫集）。作为Banach-Mazur博弈有效化的应用，我们证明该结论在有效设定下同样成立。