Functorial Boxes in String Diagrams [chapter]

Paul-André Melliès
2006 Lecture Notes in Computer Science  
String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in proof-theory (like Jean-Yves Girard's proof-nets) and in concurrency theory (like Robin Milner's bigraphs). This is not the case however, at least because string diagrams do not accomodate boxes -a key ingredient in these pictorial systems. In this short tutorial, based on our
more » ... ntal rediscovery of an idea by Robin Cockett and Robert Seely, we explain how string diagrams may be extended with a notion of functorial box to depict a functor separating an inside world (its source category) from an outside world (its target category). We expose two elementary applications of the notation: first, we characterize graphically when a faithful balanced monoidal functor F : C −→ D transports a trace operator from the category D to the category C, and we then exploit this to construct well-behaved fixpoint operators in cartesian closed categories generated by models of linear logic; second, we explain how the categorical semantics of linear logic induces that the exponential box of proof-nets decomposes as two enshrined functorial boxes.
doi:10.1007/11874683_1 fatcat:zphjcspjybch7ax6wqx7jox6vy