Abstract for: A convenient category for games and interaction

by Koslowski

Guided by the familiar construction of the category rel of relations, we first construct an order-enriched category gam . Objects are sets, and 1-cells are games, viewed as special kinds of trees. The quest for identities for the composition of arbitrary trees naturally suggests alternating trees of a specific orientation. Disjoint union of sets induces a tensor product $\otimes$ and an operation --o on gam that allow us to recover the monoidal closed category of games and strategies of interest in game theory. Since gam does not have enough maps, \ie, left adjoint 1-cells, these operations do not have nice intrinsic descriptions in gam . This leads us to consider games with explicit delay moves. To obtain the ``projection'' maps lacking in gam , we consider the Kleisli-category K induced by the functor _+1 on the category of maps in gam . Then we extend gam as to have K as category of maps. Now a satisfactory intrinsic description of the tensor product exists, which also allows us to express --o in terms of simpler operations. This construction makes clear why $\multimap$, the key to the notion of strategy, cannot be functorial on gam . Nevertheless, the composition of games may be viewed as orthogonal to the familiar composition of strategies in a common framework.