Polarized category theory, modules, and game semantics
J.R.B. Cockett, R. A. G. Seely · Theory and applications of categories · 2007
Motivated by an analysis of Abramsky-Jagadeesan games, the paper considers a categorical semantics for a polarized notion of two-player games, a semantics which has close connections with the logic of (finite cartesian) sums and products, as well as with the multiplicative structure of linear logic.In each case, the structure is polarized, in the sense that it will be modelled by two categories, one for each of two polarities, with a module structure connecting them.These are studied in considerable detail, and a comparison is made with a different notion of polarization due to Olivier Laurent: there is an adjoint connection between the two notions.Contents 1 Basic polarized games 8 2 Basic polarized game logic 12 3 Polarized categories 16 4 The logic of polarized cut and its semantics 24 5 Additive types for polarized polycategories 37 6 Linear polarized categories 55 7 Multiplicative and additive structure on AJ games 80 8 Depolarization 84 9 Exponential structure 88 10 Laurent polarized games 93 11 Concluding remarks 99Research partially supported by NSERC, Canada.Diagrams in this paper were produced with the help of the X Y -pic macros of K. Rose and R. Moore, the diagxy macros of M. Barr, and T E XCAD by G. de Montmollin.