Cauchy characterization of enriched categories

  • Ross Street


A characterization is given of those bicategories which are biequivalent to categories of modules for some suitable base. These bicategories are the correct (non elementary) notion of cosmos, which is shown to be closed under several basic constructions.


Category Theory Full Subcategory Small Category Characterization Theorem Enrich 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.


Si caratterizzano le bicategorie di moduli fra categorie arricchite, a meno di biequivalenze. Tale caratterizzazione permette di introdurre la nozione di comsos (non elementare), e di mostrare che risulta chiusa rispetto a numerose costruzioni.


  1. [1]
    Artin M., Grothendieck A. &Verdier J. L., Editors,Théorie des topos et cohomologic etale des schémas. Lecture Notes in Math. 269 (Springer, Berlin-N. Y., 1972).Google Scholar
  2. [2]
    Barr M.,Exact categories. Lecture Notes in Math. 236 (Springer, Berlin-N. Y., 1971) 1–120.CrossRefzbMATHGoogle Scholar
  3. [3]
    Bénabou J. Introduction to bicategories. Lecture Notes in Math. 47 (1967), 1–77.CrossRefGoogle Scholar
  4. [4]
    Bénabou J.,2-Dimensional limits and colimits of distributors. Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 30 (1972) 6–7.Google Scholar
  5. [5]
    Bénabou J.,Les distributeurs. Univ. Catholique de Louvain, Seminaires de Math. Pure, Rapport No. 33 (1973).Google Scholar
  6. [6]
    Betti R. andCarboni A.,Cauchy completion and the associated sheaf. Cahiers de topologie et géom. diff. 23 (1982), 243–256.MathSciNetzbMATHGoogle Scholar
  7. [7]
    Betti R. andCarboni, A.,A notion of topology for bicategories. Cahiers de topologie et géom. diff. (to appear, 1983).Google Scholar
  8. [8]
    Betti R., Carboni A., Street R. H. & Walters R. F. C.,Variation through enrichment (J. Pure & Appl. Alg. to appear).Google Scholar
  9. [9]
    Carboni A.,Categorie di relazioni. Istituto Lombardo (Rend. Sc.) A 110 (1976) 342–350MathSciNetGoogle Scholar
  10. [10]
    Dubuc E.,Adjoint triangles. Lecture Notes in Math. 61 (1968) 66–91.Google Scholar
  11. [11]
    Freyd P.,Abelian categories. (Harper and Row, New York, 1964).zbMATHGoogle Scholar
  12. [12]
    Freyd P.,On canonizing category theory, or, on functorializing model theory. (A pamphlet, University of Pennsylvania, March 1974).Google Scholar
  13. [13]
    Kelly G. M. andStreet R. H.,Review of the elements of 2-categories. Lecture Notes in Math. 420 (1974), 75–103.CrossRefMathSciNetGoogle Scholar
  14. [14]
    Lawvere F. W.,Closed categories and biclosed bicategories. Lectures at Math. Inst. Aarhus Universitet (Fall, 1971).Google Scholar
  15. [15]
    Lawvere F. W.,Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43 (1973), 135–166.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Mac Lane S.,Categories for the Working Mathematician. (Springer-Verlag, New York-Heidelberg-Berlin, 1971).zbMATHGoogle Scholar
  17. [17]
    Street R. H. The formal theory of monads. J. Pure & Appl. Algebra 2 (1972), 149–168.CrossRefMathSciNetzbMATHGoogle Scholar
  18. [18]
    Street R. H.,Two universal properties for the category of sets in the 2-category of categories. Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 30 (1972) 51–53.Google Scholar
  19. [19]
    Street R. H.,Elementary cosmoi. Lecture Notes in Math. 420 (1974), 134–180.CrossRefMathSciNetGoogle Scholar
  20. [20]
    Strett R. H. Fibrations in bicategories. Cahiers de topologie et géom. diff. 21 (1980), 111–160.Google Scholar
  21. [21]
    Street R. H.,Cosmoi of internal categories. Transactions American Math. Soc. 258 (1980), 271–318.CrossRefMathSciNetzbMATHGoogle Scholar
  22. [22]
    Street R. H.,Conspectus of variable categories. J. Pure & Appl. Algebra 21 (1981), 307–338.CrossRefMathSciNetzbMATHGoogle Scholar
  23. [23]
    Street R. H.,Enriched categories and cohomology. Quaestiones Mathematicae (to appear, 1982).Google Scholar
  24. [24]
    Street R. H.,Characterization of bicategories of stacks. Lecture Notes in Math. (Proceedings of Gummersbach Conference, 1981, to appear).Google Scholar
  25. [25]
    Street R. H. andWalters R. F. C.,Yoneda structures on 2-categories. J. Algebra 50 (1978), 350–379.CrossRefMathSciNetzbMATHGoogle Scholar
  26. [26]
    Succi-Cruclani R.,La teoria delle relazioni nello studio di categorie regolari e di categorie esatte. Riv. Math. Univ. 4 (1975) 143–158.Google Scholar
  27. [27]
    Thiébaud M.,Self-dual structure-semantics and algebraic categories. (Thesis, Dalhousie University, Halifax, August 1971).Google Scholar
  28. [28]
    Walters R. F. C.,Sheaves on sites as cauchy-complete categories. J. Pure & Appl. Algebra 24 (1982), 95–102.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag 1981

Authors and Affiliations

  • Ross Street
    • 1
  1. 1.della Macquarie UniversitySydney(Australia)

Personalised recommendations