  Pursuing a new approach to the study of infinite games in combinatorics, we
introduce the categories $\mathbf{Game}_{{A}}$ and $\mathbf{Game}_{{B}}$ and
provide novel proofs of some classical results concerning topological games
related to the duality between covering properties of $X$ and convergence
properties of $\mathrm{C}_{\mathrm {p}}(X)$ that are based on the existence of
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.
