Abstract Scalars, Loops, and Free Traced and Strongly Compact Closed Categories

  • Samson Abramsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3629)


We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures support a notion of scalar which allows quantitative aspects of physical theory to be expressed, and how the notion of strong compact closure emerges as a significant refinement of the more classical notion of compact closed category.

We then proceed to an extended discussion of free constructions for a sequence of progressively more complex kinds of structured category, culminating in the strongly compact closed case. The simple geometric and combinatorial ideas underlying these constructions are emphasized. We also discuss variations where a prescribed monoid of scalars can be ‘glued in’ to the free construction.


Tensor Product Monoidal Category Monoidal Structure Algebraic Description Symmetric Monoidal Category 
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.
    Abramsky, S., Jagadeesan, R.: New Foundations for the Geometry of Interaction. Information and Computation 111(1), 53–119 (1994); Conference version appeared in LiCS 1992 (1992) Google Scholar
  2. 2.
    Abramsky, S.: Retracing some paths in process algebra. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 1–17. Springer, Heidelberg (1996)Google Scholar
  3. 3.
    Abramsky, S., Haghverdi, E., Scott, P.J.: Geometry of Interaction and Linear Combinatory Algebras. Mathematical Structures in Computer Science 12, 625–665 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LiCS 2004), pp. 415–425. IEEE Computer Science Press, Los Alamitos (2004); (extended version at arXiv: quant-ph/0402130)Google Scholar
  5. 5.
    Abramsky, S., Coecke, B.: Abstract Physical Traces. Theory and Applications of Categories 14, 111–124 (2005)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Abramsky, S., Duncan, R.W.: Categorical Quantum Logic. In: The Proceedings of the Second International Workshop on Quantum Programming Languages (2004)Google Scholar
  7. 7.
    Barr, M.: *-autonomous Categories. Springer, Heidelberg (1979)Google Scholar
  8. 8.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Annals of Mathematics 37, 823–843 (1937)CrossRefGoogle Scholar
  9. 9.
    Coecke, B.: Delinearizing Linearity. Draft paper (2005)Google Scholar
  10. 10.
    Deligne, P.: Catégories Tannakiennes. In: The Grothendiek Festschrift, Birkhauser, vol. II, pp. 111–195 (1990)Google Scholar
  11. 11.
    Girard, J.-Y.: Linear Logic. Theoretical Computer Science 50(1), 1–102 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Girard, J.-Y.: Geometry of Interaction I: Interpretation of System F. In: Ferro, R., et al. (eds.) Logic Colloquium 1988, pp. 221–260. North-Holland, Amsterdam (1989)Google Scholar
  13. 13.
    Hyland, M.: Personal communication (July 2004)Google Scholar
  14. 14.
    Joyal, A., Street, R., Verity, D.: Traced monoidal categories. Math. Proc. Camb. Phil. Soc. 119, 447–468 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kassel, C.: Quantum Groups. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  16. 16.
    Katis, P., Sabadini, N., Walters, R.F.C.: Feedback, trace and fixed point semantics. In: Proceedings of FICS 2001: Workshop on Fixed Points in Computer Science (2001), Available at
  17. 17.
    Kelly, G.M.: Many-variable functorial calculus I. Lecture Notes in Mathematics, vol. 281, pp. 66–105. Springer, Heidelberg (1972)Google Scholar
  18. 18.
    Kelly, G.M.: An abstract approach to coherence. Lecture Notes in Mathematics, vol. 281, pp. 106–147. Springer, Heidelberg (1972)Google Scholar
  19. 19.
    Kelly, G.M., Laplaza, M.L.: Coherence for compact closed categories. Journal of Pure and Applied Algebra 19, 193–213 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)zbMATHGoogle Scholar
  21. 21.
    Pavlovic, D.: A semantical approach to equilibria, adaptation and evolution. Unpublished manuscript (November 2004)Google Scholar
  22. 22.
    Penrose, R.: Applications of negative-dimensional tensors. In: Welsh, D.J. (ed.) Combinatorial Mathematics and Its Applications, pp. 221–244. Academic Press, London (1971)Google Scholar
  23. 23.
    Rivano, N.S.: Catégories Tannakiennes. Springer, Heidelberg (1972)zbMATHGoogle Scholar
  24. 24.
    Selinger, P.: Towards a quantum programming language. Mathematical Structures in Computer Science 14(4), 527–586 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Selinger, P.: Dagger compact closed categories and completely positive maps. In: Proceedings of the 3rd International Workshop on Quantum Programming Languages (2005) (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Samson Abramsky
    • 1
  1. 1.Computing LaboratoryOxford UniversityOxfordU.K

Personalised recommendations