Extensions of Innocence

How do strategies compose and why do they compose associatively? Early
accounts of game semantics treated these questions in a direct
fashion. However both the discovery of implementations by abstract
machines, and experience with abstract games suggest a range of other
points of view.
I shall outline two further categorical contexts, one related to
simple strategies, and one dealing with innocent strategies. I shall
indicate one might develop these ideas to extend the notion of
innocence.