Skip to main content

Fibrations and partial products in a 2-category

Applied Categorical Structures volume 1, pages 141–179 (1993)Cite this 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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  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.

  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. A. Carboni and P. T. Johnstone, ‘Connected limits, familial representability and Artin glueing’, in preparation.

  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. R. Dyckhoff, ‘Total reflections, partial products and hereditary factorisations’,Topology Appl. 17 (1984), 101–113.

    Google Scholar 

  6. R. Dyckhoff and W. Tholen, ‘Exponentiable morphisms, partial products and pullback complements’,J. Pure Appl. Alg. 49 (1987), 103–116.

    Google Scholar 

  7. J. W. Gray, ‘Fibred and cofibred categories’, inProceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer-Verlag, 1966, 21–83.

  8. J. W. Gray,Formal Category Theory: Adjointness for 2-Categories. Lecture Notes in Math. vol. 391, Springer-Verlag, 1974.

  9. A. Grothendieck, ‘Catégories fibrées et descente’, Séminaire de Géométrie Algébrique de l'I.H.E.S., 1961.

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

  11. P. T. Johnstone, ‘The Gleason cover of a topos, II’,J. Pure Appl. Alg. 22 (1981), 229–247.

    Google Scholar 

  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.

  13. P. T. Johnstone, ‘Variations on the bagdomain theme’, in preparation.

  14. P. T. Johnstone, ‘Fibrations and pro-arrow equipment’, in preparation.

  15. P. T. Johnstone and A. Joyal, ‘Continuous categories and exponentiable toposes’,J. Pure Appl. Alg. 25 (1982), 255–296.

    Google Scholar 

  16. P. T. Johnstone and I. Moerdijk, ‘Local maps of toposes’,Proc. Lond. Math. Soc. (3)58 (1989), 281–305.

    Google Scholar 

  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.

  18. A. Joyal and R. Street, ‘Pullbacks equivalent to pseudopullbacks’, Macquarie Mathematics Report no. 92-092, 1992.

  19. G. M. Kelly, ‘On clubs and doctrines’, inCategory Seminar, Sydney 1972/73, Lecture Notes in Math. vol. 420, Springer-Verlag, 1974, 181–256.

  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.

  21. B. A. Pasynkov, ‘Partial topological products’,Trans. Moscow Math. Soc. 13 (1965), 153–272.

    Google Scholar 

  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. 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. R. D. Rosebrugh and R. J. Wood, ‘Proarrows and cofibrations’,J. Pure Appl. Alg. 53 (1988), 271–296.

    Google Scholar 

  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.

  26. R. Street, ‘Fibrations in bicategories’,Cahiers Top. Géom. Diff. 21 (1980), 111–160.

    Google Scholar 

  27. R. Street, ‘Conspectus of variable categories’,J. Pure Appl. Alg. 21 (1981), 307–338.

    Google Scholar 

  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.

  29. R. J. Wood, ‘Abstract proarrows I’,Cahiers Top. Géom. Diff. 23 (1982), 279–290.

    Google Scholar 

  30. G. C. Wraith, ‘Artin glueing’,J. Pure Appl. Alg. 4 (1974), 345–348.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Cambridge, 16 Mill Lane, CB2 15B, Cambridge, U.K.

    P. T. Johnstone

Authors
  1. P. T. Johnstone
    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

Johnstone, P.T. Fibrations and partial products in a 2-category. Appl Categor Struct 1, 141–179 (1993). https://doi.org/10.1007/BF00880041

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00880041

Mathematics Subject Classifications (1991)

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

Key words

  • Fibration
  • partial product
  • 2-category