Functorial Boxes in String Diagrams

  • Paul-André Melliès
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)


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.


Categorical Semantic Monoidal Category Intuitionistic Logic Linear Logic Graphical Notation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Amadio, R., Curien, P.-L.: Domains and Lambda-Calculi. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  2. 2.
    Barber, A., Gardner, P., Hasegawa, M., Plotkin, G.: From action calculi to linear logic. In: Nielsen, M. (ed.) CSL 1997. LNCS, vol. 1414. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Barber, A.: Linear Type Theories, Semantics and Action Calculi. PhD Thesis of the University of Edinburgh. LFCS Technical Report CS-LFCS-97-371 (1997)Google Scholar
  4. 4.
    Bénabou, J.: Introduction to bicategories. In: Reports of the Midwest Category Seminar. Lecture Notes in Mathematics, vol. 47. Springer, Heidelberg (1967)CrossRefGoogle Scholar
  5. 5.
    Benton, N.: A Mixed Linear and Non-Linear Logic: Proofs, Terms and Models. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  6. 6.
    Benton, N., Bierman, G., de Paiva, V., Hyland, M.: Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge (1992)Google Scholar
  7. 7.
    Berry, G.: Stable models of typed lambda-calculi. In: Ausiello, G., Böhm, C. (eds.) ICALP 1978. LNCS, vol. 62. Springer, Heidelberg (1978)Google Scholar
  8. 8.
    G. Bierman. On intuitionistic linear logic. PhD Thesis. University of Cambridge Computer Laboratory (December 1993)Google Scholar
  9. 9.
    Bierman, G.: What is a categorical model of intuitionistic linear logic? In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 73–93. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  10. 10.
    Blackwell, H., Kelly, M., Power, A.J.: Two dimensional monad theory. Journal of Pure and Applied Algebra 59, 1–41 (1989)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Blute, R., Cockett, R., Seely, R.: The logic of linear functors. Mathematical Structures in Computer Science 12(4), 513–539 (2002)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Blute, R., Cockett, R., Seely, R., Trimble, T.: Natural Deduction and Coherence for Weakly Distributive Categories. Journal of Pure and Applied Algebra 113(3), 229–296 (1996)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Burroni, A.: Higher Dimensional Word Problem. In: Curien, P.-L., Pitt, D.H., Pitts, A.M., Poigné, A., Rydeheard, D.E., Abramsky, S. (eds.) CTCS 1991. LNCS, vol. 530. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  14. 14.
    Cockett, R., Seely, R.: Linearly Distributive Categories. Journal of Pure and Applied Algebra 114(2), 133–173 (1997)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Cockett, R., Seely, R.: Linear Distributive Functors. The Barrfestschrift, Journal of Pure and Applied Algebra 143(1-3) (November 10, 1999)Google Scholar
  16. 16.
    Day, B.J., Street, R.: Quantum categories, star autonomy, and quantum groupoids. Galois Theory, Hopf Algebras, and Semiabelian Categories. Fields Institute Communications 43 (American Math. Soc. 2004), 187–226 (2004)Google Scholar
  17. 17.
    Gentzen, G.: Investigations into logical deduction (1934); An english translation appears. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, North-Holland, Amsterdam (1969)Google Scholar
  18. 18.
    Girard, J.-Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Girard, J.-Y.: Linear logic: its syntax and semantics. In: Advances in Linear Logic. London Mathematical Society Lecture Note Series, vol. 222, pp. 1–42. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  20. 20.
    Hasegawa, M.: Recursion from cyclic sharing: traced monoidal categories and models of cyclic lambda-calculi. In: de Groote, P., Hindley, J.R. (eds.) TLCA 1997. LNCS, vol. 1210. Springer, Heidelberg (1997)Google Scholar
  21. 21.
    Hasegawa, R.: Two applications of analytic functors. Theoretical Computer Science 272, 113–175 (2002)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Hermida, C., Power, J.: Fibrational control structures. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 117–129. Springer, Heidelberg (1995)Google Scholar
  23. 23.
    Hyland, M., Schalk, A.: Glueing and orthogonality for models of linear logic. Theoretical Computer Science 294(1/2), 183–231 (2003)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Im, G.B., Kelly, M.: A universal property of the convolution monoidal structure. J. Pure Appl. Algebra 43, 75–88 (1986)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Kelly, M.: Doctrinal adjunction. Lecture Notes in Math., vol. 420, pp. 257–280 (1974)Google Scholar
  26. 26.
    Joyal, A.: Remarques sur la théorie des jeux à deux personnes. Gazette des Sciences Mathématiques du Québec 1(4), 46–52 (1977)Google Scholar
  27. 27.
    Joyal, A., Street, R.: Braided Tensor Categories. Advances in Mathematics 102, 20–78 (1993)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Joyal, A., Street, R.: The geometry of tensor calculus, I. Advances in Mathematics 88, 55–112 (1991)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Camb. Phil. Soc. 119, 447–468 (1996)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Lack, S.: Limits for lax morphisms. Applied Categorical Structures 13(3), 189–203 (2005)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Lafont, Y.: Logiques, catégories et machines. PhD thesis, Université Paris 7 (1988)Google Scholar
  32. 32.
    Lafont, Y.: From Proof Nets to Interaction Nets. In: Advances in Linear Logic. London Mathematical Society Lecture Note Series, vol. 222, pp. 225–247. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  33. 33.
    Lamarche, F.: Sequentiality, games and linear logic (unpublished manuscript, 1992)Google Scholar
  34. 34.
    Lambek, J., Scott, P.: Introduction to Higher Order Categorical Logic. Cambridge Studies in Advanced Mathematics, vol. 7. Cambridge University Press, Cambridge (1986)MATHGoogle Scholar
  35. 35.
    Lawvere, F.W.: Ordinal sums and equational doctrines. Springer Lecture Notes in Mathematics, vol. 80, pp. 141–155. Springer, Berlin (1969)Google Scholar
  36. 36.
    Lane, S.M.: Categories for the working mathematician. Graduate Texts in Mathematics, 2nd edn., vol. 5. Springer, Heidelberg (1998)MATHGoogle Scholar
  37. 37.
    Maietti, M., Maneggia, P., de Paiva, V., Ritter, E.: Relating categorical semantics for intuitionistic linear logic. Applied Categorical Structures 13(1), 1–36 (2005)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Melliès, P.-A.: Typed lambda-calculi with explicit substitutions not terminate. In: Dezani-Ciancaglini, M., Plotkin, G. (eds.) TLCA 1995. LNCS, vol. 902, pp. 328–334. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  39. 39.
    Melliès, P.-A.: Axiomatic Rewriting 4: a stability theorem in rewriting theory. In: Proceedings of Logic in Computer Science 1998. IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  40. 40.
    Melliès, P.-A.: Categorical semantics of linear logic: a survey. Panoramas et Synthèses, Société Mathématique de France (to appear, 2007)Google Scholar
  41. 41.
    Milner, R.: Pure bigraphs: structure and dynamics. Information and Computation 204(1) (January 2006)Google Scholar
  42. 42.
    Pavlovic, D.: Categorical logic of names and abstraction in action calculi. Mathematical Structures in Computer Science 7(6), 619–637 (1997)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Penrose, R.: Applications of negative dimensional tensors. In: Welsh, D.J.A. (ed.) Combinatorial Mathematics and its Applications, pp. 221–244. Academic Press, New York (1971)Google Scholar
  44. 44.
    Penrose, R.: Spinors and Space-Time, vol. 1, pp. 68–71, 423-434. Cambridge University Press, Cambridge (1984)CrossRefMATHGoogle Scholar
  45. 45.
    Seely, R.: Linear logic, *-autonomous categories and cofree coalgebras. Applications of categories in logic and computer science, Contemporary Mathematics, 92 (1989)Google Scholar
  46. 46.
    Schanuel, S., Street, R.: The free adjunction. Cahiers topologie et géométrie différentielle catégoriques 27, 81–83 (1986)MathSciNetGoogle Scholar
  47. 47.
    Street, R.: Limits indexed by category-valued 2-functors. J. Pure Appl. Algebra 8, 149–181 (1976)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Street, R.: Functorial calculus in monoidal bicategories. Applied Categorical Structures 11, 219–227 (2003)CrossRefMathSciNetGoogle Scholar
  49. 49.
    Tabareau, N.: De l’opérateur de trace dans les jeux de Conway. Mémoire de Master 2. Master Parisien de Recherche en Informatique, Université Paris 7 (September 2005)Google Scholar

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 

Personalised recommendations