We present a simple construction of an order-enriched category gam that simultaneously dualizes and parallels the familiar construction of the category rel of relations. Objects of gam are sets, and arrows are games, viewed as special kinds of trees. The quest for identities for the composition of trees naturally leads to the consideration of alternating sequences and games of a specific polarity. gam may be viewed as a canonical extension of rel , and just as for rel , the maps in gam admit a nice characterization. Disjoint union of sets induces a special tensor product on gam that allows us to recover the monoidal closed category of games and strategies of interest in game theory. If we allow games with explicit delay moves, the categorical description of the structure that leads to the monoidal closed category is even more satisfying. In particular, we then obtain an explicit involution. 0 Introduction People who study games from a mathematical perspective (e.g., Blass, Abramsky, ..
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