Abstract for: A convenient category for games and interaction
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.