International Workshop on Computer Science Logic

CSL 2006: Computer Science Logic pp 1-30

Functorial Boxes in String Diagrams

  • Paul-André Melliès
Conference paper

DOI: 10.1007/11874683_1

Volume 4207 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Melliès PA. (2006) Functorial Boxes in String Diagrams. In: Ésik Z. (eds) Computer Science Logic. CSL 2006. Lecture Notes in Computer Science, vol 4207. Springer, Berlin, Heidelberg

Abstract

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 depicting a functor transporting an inside world (its source category) to 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 \(\longrightarrow\)\(\mathbb{D}\) transports a trace operator from the category \(\mathbb{D}\) to the category ℂ, and 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 boxes.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Paul-André Melliès
    • 1
  1. 1.Equipe Preuves, Programmes, Systèmes, CNRS — Université Paris 7 Denis Diderot