Applied Categorical Structures

, Volume 10, Issue 2, pp 157–172 | Cite as

Localization of Universal Problems. Local Colimits

  • Andrée C. Ehresmann


The notion of the root of the category, which is a minimal (in a precise sense) weakly coreflective subcategory, is introduced in view of defining ‘local’ solutions of universal problems: If U is a functor from C′ to C and c an object of C, the root of the comma-category c|U is called a U-universal root generated by c; when it exists, it is unique (up to isomorphism) and determines a particular form of the locally free diagrams defined by Guitart and Lair. In this case, the analogue of an adjoint functor is an adjoint-root functor of U, taking its values in the category of pro-objects of C′. Local colimits are obtained if U is the insertion from a category into its category of ind-objects; they generalize Diers' multicolimits. Applications to posets and Galois theory are given.

category universal problem colimit Galois theory poset 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ageron, P.: Catégories accessibles à produits fibrés, Diagrammes 36 (1996).Google Scholar
  2. 2.
    Barr, M.: Abstract Galois theory, J. Pure Appl. Algebra 19 (1980), 21–42.Google Scholar
  3. 3.
    Deleanu, A. and Hilton, P.: Borsuk shape and Grothendieck categories of pro-objects, Math. Proc. Cambridge 79 (1976), 473–482.Google Scholar
  4. 4.
    Diers, Y.: Catégories localisables, Thèse Université Paris VI, dy1971.Google Scholar
  5. 5.
    Diers, Y.: Catégories multialgébriques galoisiennes, Cahiers Topologie Géom. Différentielle Catégoriques XXXIII(1) (1992), 55–69.Google Scholar
  6. 6.
    Duskin, J.: Pro-objects (d'après Verdier), Séminaire Heidelberg-Strasbourg 1966-67, Exposé 6.Google Scholar
  7. 7.
    Ehresmann, A. C.: Comments on part IV-1. In: Charles Ehresmann: Oeuvres complètes et commentées IV-1, Amiens, 1981, p. 370.Google Scholar
  8. 8.
    Ehresmann, A. C. and Vanbremeersch, J.-P.: Hierarchical evolutive systems: A mathematical model for complex systems, Bull. Math. Biol. 49(1) (1987), 13–50. And Multiplicity principle and emergence in memory evolutive systems, SAMS 26 (1997), 81-117.Google Scholar
  9. 9.
    Ehresmann, C.: Groupoïdes sous-inductifs, Ann. Inst. Fourier (Grenoble) 13(2) (1963), 1–60;reprinted in Charles Ehresmann: Oeuvres complètes et commentées, partie II-1, Amiens, 1982,p. 257.Google Scholar
  10. 10.
    Grothendieck, A.: Revêtements étales et groupe fondamental, S.G.A. 1, Lecture Notes in Math. 224, Springer, 1971.Google Scholar
  11. 11.
    Guitart, R.: On the geometry of computations I and II, Cahiers Topologie Géom. Différentielle Catégorique XXVII(4) (1988), 107–136; XXIX(4) (1990), 297-326.Google Scholar
  12. 12.
    Guitart, R. and Lair, C.: Calcul syntaxique des modèles et calcul des formules internes, Diagrammes 4 (1980).Google Scholar
  13. 13.
    Lair, C.: Diagrammes localement libres. Extensions de corps et théorie de Galois, Diagrammes 10 (1983).Google Scholar
  14. 14.
    Lair, C.: Sur le profil d'esquissabilité des catégories modelables possédant les noyaux, Diagrammes 36 (1996).Google Scholar
  15. 15.
    Mac Lane, S.: Categories for the Working Mathematician, Springer, 1971.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Andrée C. Ehresmann
    • 1
  1. 1.Faculté de Mathématiques et d'InformatiqueAmiensFrance

Personalised recommendations