Game theory has applications to logic, philosophy, economics, political science, biology, computer sciences and ... category theory! Conway's games are the objects of a category in which the maps are winning communication strategies; the category is symmetric monoidal closed and compact. Whitman's solution of the word problem for free lattices can be described in terms of game theory. The theorem can be extended to free bicomplete categories. I will discuss potential applications to the construction of new categories of games.
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