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Fibered categories and the foundations of naive category theory

  • Jean Bénabou (a1)


Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives, of course related.

(0.1) Noncontradiction: Namely, to provide a formal frame rich enough so that all the actual activity in the domain can be carried out within this frame, and consistent, or at least relatively consistent with a well-established and “safe” theory, e.g. Zermelo-Frankel (ZF).

(0.2) Adequacy, in the following, nontechnical sense:

(i) The basic notions must be simple enough to make transparent the syntactic structures involved.

(ii) The translation between the formal language and the usual language must be, or very quickly become, obvious. This implies in particular that the terminology and notations in the formal system should be identical, or very similar, to the current ones. Although this may seem minor, it is in fact very important.

(iii) “Foundations” can only be “foundations of a given domain at a given moment”, therefore the frame should be easily adaptable to extensions or generalizations of the domain, and, even better, in view of (i), it should suggest how to find meaningful generalizations.

(iv) Sometimes (ii) and (iii) can be incompatible because the current notations are not adapted to a more general situation. A compromise is then necessary. Usually when the tradition is very strong (ii) is predominant, but this causes some incoherence for the notations in the more general case (e.g. the notation f(x) for the value of a function f at x obliges one, in category theory, to denote the composition of arrows (f, g) → g∘f, and all attempts to change this notation have, so far, failed).



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[Bé.1] Bénabou, J., Lectures, Université de Montréal, Montréal, 1974 (unpublished).
[Bé.2] Bénabou, J., Théories relatives à un corpus, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris), Séries A et B, vol. 281 (1975), pp. A831A834.
[Bé.3] Bénabou, J., Fibrations petites et localement petites, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (Paris), Séries A et B, vol. 281 (1975), pp. A897A900.
[Bé.4] Bénabou, J., Des catégories fibrées (Book, in preparation; some parts are available in preprint form).
[Cel] Celeyrette, J., Fibrations et extensions de Kan, Thèse, Université Paris-Nord, Paris, 1975.
[Gi] Giraud, J., Cohomologie non abélienne, Springer-Verlag, Berlin, 1971.
[Gr] Grothendieck, A., Catégories fibrées et descente, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois-Marie 1960/61 (SGA 1), Exposé VI, 3rd ed., Institut des Hautes Études Scientifiques, Paris, 1963; reprint, Lecture Notes in Mathematics, vol. 224, Springer-Verlag, Berlin, 1971, pp. 145194.
[Joh] Johnstone, P. T., Topos theory, Academic Press, London, 1977.
[Joh-Par] Johnstone, P. T. and Paré, R. (editors), Indexed categories and their applications, Lecture Notes in Mathematics, vol. 661, Springer-Verlag, Berlin, 1978.
[Mo] Moens, , Fibrations géométiriques et théorème de Giraud, Thèse, Louvain-la-Neuve, 1982. (Abriged version, to appear.)
[Par-Sch] Paré, R. and Schumacher, D., Abstract families and the adjoint functor theorems, in [Joh-Par, pp. 1–125]
[SGA] Artin, M., Grothendieck, A. and Verdier, J. L. (editors), Théorie des topos et cohomologie étale des schémas, Séminaire de Géométrie Algébrique du Bois-Marie 1963/64 (SGA 4), Tomes 1, 2, rev. ed., Lecture Notes in Mathematics, vols. 269, 270, Springer-Verlag, Berlin, 1972.
[TD] Grothendieck, A., Technique de descente et théorèmes d'existence en géométrie algébrique. I, Séminaire Bourbaki 1959/60, Exposé 190, Secrétariat Mathématique, Paris, 1969.

Fibered categories and the foundations of naive category theory

  • Jean Bénabou (a1)


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