The characterization of second-order type isomorphisms is a purely
syntactical problem that we propose to study under the enlightenment of game
semantics. We study this question in the case of second-order
λμ-calculus, which can be seen as an extension of system F to
classical logic, and for which we define a categorical framework: control
hyperdoctrines. Our game model of λμ-calculus is based on polymorphic
arenas (closely related to Hughes' hyperforests) which evolve during the play
(following the ideas of Murawski-Ong). We show that type isomorphisms coincide
with the "equality" on arenas associated with types. Finally we deduce the
equational characterization of type isomorphisms from this equality. We also
recover from the same model Roberto Di Cosmo's characterization of type
isomorphisms for system F. This approach leads to a geometrical comprehension
on the question of second order type isomorphisms, which can be easily extended
to some other polymorphic calculi including additional programming features.Comment: accepted by Annals of Pure and Applied Logic, Special Issue on Game
Semantic