Skip to main content
SpringerLink
Log in

About И-quantifiers

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

Gabbay and Pitts observed that the Fraenkel–Mostowski model of set-theory supports useful notions of “name-abstraction” and “fresh-name”. In order to understand their work in a more general setting we introduce the notions of И-units and И-relations in a regular category D. A И-relation is given by a functor A # (-):D→D and we show that in the case that D is a topos then A # (-) has a right adjoint [A](-) that can be thought of as an object of abstractions. We also explore the existence of a right adjoint to [A](-) and relate it to the “name swapping” operations considered as fundamental by Gabbay and Pitts. We present many examples of categories where this notions occur and we relate the results here with Pitts' Nominal Logic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barr, M., Grillet, P. A. and van Osdol, D. H.: Exact Categories and Categories of Sheaves, Lecture Notes in Mathematics 236, Springer-Verlag, New York, 1971.

    Google Scholar 

  2. Bergeron, F., Labelle, G. and Leroux, P.: Combinatorial Species and Tree-Like Structures, Vol. 67, Cambridge University Press, 1998.

  3. Borceux, F.: Handbook of Categorical Algebra 2, Encyclopedia of Mathematics and its Applications 51, Cambridge University Press, 1994.

  4. Day, B.: On closed categories of functors, in Reports of the Midwest Category Seminar, Lecture Notes in Mathematics 137, Springer-Verlag, New York, 1970, pp. 1–38.

    Google Scholar 

  5. Eilenberg, S. and Kelly, G. M.: Closed categories, in Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer, New York, 1966, pp. 421–562.

    Google Scholar 

  6. Fiore, M.: Notes on combinatorial functors, Draft, January 2001.

  7. Fiore, M., Plotkin, G. D. and Turi, D.: Abstract syntax and variable binding, in 14th Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, Washington, 1999.

    Google Scholar 

  8. Gabbay, J. and Pitts, A.M.: A new approach to abstract syntax with variable binding, Formal Aspects of Computing 13 (2002), 341–363. Special issue in honour of Rod Burstall.

    Google Scholar 

  9. Hofmann, M.: Semantical analysis of higher order abstract syntax, in 14th Annual Symposium on Logic in Computer Science, IEEE Computer Society Press, Washington, 1999.

    Google Scholar 

  10. Johnstone, P. T.: Topos Theory, Academic Press, 1977.

  11. Joyal, A.: Une theorie combinatoire des séries formelles, Advances in Mathematics 42 (1981), 1–82.

    Google Scholar 

  12. Kock, A. and Reyes, G. E.: Aspects of fractional exponent functors, Theory and Applications of Categories 5(10) (1999), 251–265.

    Google Scholar 

  13. Lawvere, F. W.: Adjoints in and among bicategories, in Logic and Algebra, Lecture Notes in Pure and Applied Algebra 180, Marcel Dekker, Inc., 1996, pp. 181–189. Proceedings of the 1994 Siena conference in memory of Roberto Magari.

  14. Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Mathematics, Springer-Verlag, New York, 1971.

    Google Scholar 

  15. Mac Lane, S. and Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Universitext, Springer-Verlag, 1992.

  16. Nava, O. and Rota, G.-C.: Plethysm, categories and combinatorics, Advances in Mathematics 58 (1985), 61–88.

    Google Scholar 

  17. Pitts, A. M.: Categorical logic, in S. Abramsky, D. M. Gabbay and T. S. E. Maibaum (eds), Handbook of Logic in Computer Science, Volume 5. Algebraic and Logical Structures, Oxford University Press, 2000, Chapter 2.

  18. Pitts, A. M.: Nominal logic: A first order theory of names and binding, in N. Kobayashi and B. C. Pierce (eds), Fourth International Symposium on Theoretical Aspects of Computer Software (TACS 2001), Sendai, Japan, Lecture Notes in Computer Science, Springer-Verlag, Berlin, 2001.

    Google Scholar 

  19. Pitts, A. M. and Gabbay, M. J.: A metalanguage for programming with bound names modulo renaming, in R. Backhouse and J. N. Oliviera (eds), Mathematics of Program Construction. Proceedings of the 5th International Conference, MPC2000, Ponte de Lima, Portugal, July 2000, Lecture Notes in Computer Science 1837, Springer-Verlag, New York, 2000, pp. 230–255.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LIFIA, Universidad Nacional de La Plata, calle 50 y 115, Primer piso, (1900), La Plata, Argentina

    Matías Menni

Authors
  1. Matías Menni
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Menni, M. About И-quantifiers. Applied Categorical Structures 11, 421–445 (2003). https://doi.org/10.1023/A:1025750816098

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1025750816098

Search

Navigation