Functorial Boxes in String Diagrams

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 accidental 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. ∗Invited paper at the conference Computer Science Logic 2006 in Szeged, Hungary. To appear in the proceedings of the conference. c © Springer Verlag. This research was partially supported by the ANR Project INVAL “Invariants algébriques des systèmes informatiques”.