Advertisement

Applied Categorical Structures

, Volume 1, Issue 2, pp 141–179 | Cite as

Fibrations and partial products in a 2-category

  • P. T. Johnstone
Article

Abstract

We introduce a new intrinsic definition of fibrations in a 2-category, and show how it may be used (in conjunction with a suitable limit-colimit commutation condition) to define a 2-categorical version of the notion of partial product. We use these notions to show that partial products exist for all fibrations in the 2-category of (small) categories, and to identify the fibrations in the 2-category of toposes and geometric morphisms.

Mathematics Subject Classifications (1991)

Primary 18D05, 18D30 secondary 18A30, 18B25 

Key words

Fibration partial product 2-category 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Bénabou, ‘Some remarks on free monoids in a topos’, inCategory Theory, Como 1990, Lecture Notes in Math. vol. 1488, Springer-Verlag, 1991, 20–29.Google Scholar
  2. 2.
    G. J. Bird, G. M. Kelly, A. J. Power, and R. H. Street, ‘Flexible limits for 2-categories’,J. Pure Appl. Alg. 61 (1989), 1–27.Google Scholar
  3. 3.
    A. Carboni and P. T. Johnstone, ‘Connected limits, familial representability and Artin glueing’, in preparation.Google Scholar
  4. 4.
    F. Conduché, ‘Au sujet de l'existence d'adjoints à droite aux foncteurs “image réciproque” dans la catégorie des catégories’,C. R. Acad. Sci. Paris 275 (1972), A891–894.Google Scholar
  5. 5.
    R. Dyckhoff, ‘Total reflections, partial products and hereditary factorisations’,Topology Appl. 17 (1984), 101–113.Google Scholar
  6. 6.
    R. Dyckhoff and W. Tholen, ‘Exponentiable morphisms, partial products and pullback complements’,J. Pure Appl. Alg. 49 (1987), 103–116.Google Scholar
  7. 7.
    J. W. Gray, ‘Fibred and cofibred categories’, inProceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer-Verlag, 1966, 21–83.Google Scholar
  8. 8.
    J. W. Gray,Formal Category Theory: Adjointness for 2-Categories. Lecture Notes in Math. vol. 391, Springer-Verlag, 1974.Google Scholar
  9. 9.
    A. Grothendieck, ‘Catégories fibrées et descente’, Séminaire de Géométrie Algébrique de l'I.H.E.S., 1961.Google Scholar
  10. 10.
    P. T. Johnstone,Topos Theory, Academic Press, 1977.Google Scholar
  11. 11.
    P. T. Johnstone, ‘The Gleason cover of a topos, II’,J. Pure Appl. Alg. 22 (1981), 229–247.Google Scholar
  12. 12.
    P. T. Johnstone, ‘Partial products, bagdomains and hyperlocal toposes’, inApplications of Categories in Computer Science, L.M.S. Lecture Notes Series No. 177, Cambridge University Press, 1992, 315–339.Google Scholar
  13. 13.
    P. T. Johnstone, ‘Variations on the bagdomain theme’, in preparation.Google Scholar
  14. 14.
    P. T. Johnstone, ‘Fibrations and pro-arrow equipment’, in preparation.Google Scholar
  15. 15.
    P. T. Johnstone and A. Joyal, ‘Continuous categories and exponentiable toposes’,J. Pure Appl. Alg. 25 (1982), 255–296.Google Scholar
  16. 16.
    P. T. Johnstone and I. Moerdijk, ‘Local maps of toposes’,Proc. Lond. Math. Soc. (3)58 (1989), 281–305.Google Scholar
  17. 17.
    P. T. Johnstone and G. C. Wraith, ‘Algebraic theories in toposes’, inIndexed Categories and Their Applications, Lecture Notes in Math. vol. 661, Springer-Verlag, 1978, 141–242.Google Scholar
  18. 18.
    A. Joyal and R. Street, ‘Pullbacks equivalent to pseudopullbacks’, Macquarie Mathematics Report no. 92-092, 1992.Google Scholar
  19. 19.
    G. M. Kelly, ‘On clubs and doctrines’, inCategory Seminar, Sydney 1972/73, Lecture Notes in Math. vol. 420, Springer-Verlag, 1974, 181–256.Google Scholar
  20. 20.
    G. M. Kelly, ‘On clubs and data type constructors’, inApplications of Categories in Computer Science, L.M.S. Lecture Notes Series No. 177, Cambridge University Press, 1992, 163–190.Google Scholar
  21. 21.
    B. A. Pasynkov, ‘Partial topological products’,Trans. Moscow Math. Soc. 13 (1965), 153–272.Google Scholar
  22. 22.
    R. D. Rosebrugh and R. J. Wood, ‘Cofibrations in the bicategory of topoi’,J. Pure Appl. Alg. 32 (1984), 71–94.Google Scholar
  23. 23.
    R. D. Rosebrugh and R. J. Wood, ‘Cofibrations II: left exact right actions and composition of gamuts’,J. Pure Appl. Alg. 39 (1986), 283–300.Google Scholar
  24. 24.
    R. D. Rosebrugh and R. J. Wood, ‘Proarrows and cofibrations’,J. Pure Appl. Alg. 53 (1988), 271–296.Google Scholar
  25. 25.
    R. Street, ‘Fibrations and Yoneda's lemma in a 2-category’, inCategory Seminar, Sydney 1972/73, Lecture Notes in Math. vol. 420, Springer-Verlag, 1974, 104–133.Google Scholar
  26. 26.
    R. Street, ‘Fibrations in bicategories’,Cahiers Top. Géom. Diff. 21 (1980), 111–160.Google Scholar
  27. 27.
    R. Street, ‘Conspectus of variable categories’,J. Pure Appl. Alg. 21 (1981), 307–338.Google Scholar
  28. 28.
    S. J. Vickers, ‘Geometric theories and databases’, inApplications of Categories in Computer Science, L.M.S. Lecture Notes Series No. 177, Cambridge University Press, 1992, 288–314.Google Scholar
  29. 29.
    R. J. Wood, ‘Abstract proarrows I’,Cahiers Top. Géom. Diff. 23 (1982), 279–290.Google Scholar
  30. 30.
    G. C. Wraith, ‘Artin glueing’,J. Pure Appl. Alg. 4 (1974), 345–348.Google Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • P. T. Johnstone
    • 1
  1. 1.Department of Pure MathematicsUniversity of CambridgeCambridgeU.K.

Personalised recommendations

Cite article

Buy options