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Fibered categories and the foundations of naive category theory. (English) Zbl 0564.18001
This is an important paper, which the reviewer hopes that any intending author on categorical logic or on the foundations of mathematics will read carefully. The author stresses that this is not a mathematical paper; the reader will nevertheless have to have quite a technical mathematical background to appreciate the force of the argument, and to understand what is at stake.
It is also unusually frank. Some might condemn it as contentious. Why this should be so is a complicated story. The reader should not make the mistake of dismissing it as mere polemic; the substance of what the author has to say deserves better than that, though the causes of the polemical aspect are worth learning from too.
When Lawvere posed the question of how to characterize the category of sets and functions in categorical terms, he opened the door to the possibility of founding mathematics in other than set-theoretical terms. Sets are to be objects in categories satisfying suitable properties. For such a program it is of course necessary to have a notion of category, and of categorical properties, that presupposes no set-theory. The trouble is that ”naive category theory” started out with no such high aim, so that a great many constructions, over which hands are confidently waved, rest upon shaky foundations or are plain wrong. A catechism of such errors is described in this article.
Some idea of the contents of the article can be had from the titles of the section headings: 0. Introduction. - 1. The first observations about naive category theory. - 2. The usual foundations. - 3. The basic notions of our formal system. - 4. What will the theory of ”categories” look like? - 5. Some ”paradoxes” in category theory. - 6. Formal definability. - 7. Definability and representability, or how to get rid of set theories. - 8. Equality in category theory. - 9. What is a ”category”, or a ”category” with equality of objects? - 10. ”Categories” and fibrations. - 11. Forgetting about foundations. - 12. Appendix. Fibrations and indexed categories. - Short annotated glossary.
In 1970 the author realized the possibility of using fibred categories as a basis for an approach to foundations. He is engaged on writing a book ”Des categories fibrees” in which we have been promised such an approach. Meanwhile two articles have appeared in the Comptes Rendus in 1975 [C. R. Acad. Sci., Paris, Sér. A 281, 831-834, 897-900 (1975; Zbl 0349.18005 and Zbl 0349.18006)]. It is unfortunate that many category theorists have been put off the topic of fibred categories, perhaps because an important exegesis, J. Giraud’s ”Cohomologie nonabélienne” (1971; Zbl 0226.14011), is indigestible. Penon’s locally internal categories and Pare’s indexed categories are half-measure solutions to the problems that they are intended to address, as the author makes clear forcefully. If only his own work had been publicized more widely than in Comptes Rendus, perhaps we would have been spared the false starts and painful recriminations. In any case, such speculations are unimportant. What matters is that the author has put his finger on a weak spot in naive category theory, and offers an elegant tool to repair the damage. The sooner ”Des categories fibrees” appears, the better.
Reviewer: G.C.Wraith

MSC:
18A15 Foundations, relations to logic and deductive systems
03G30 Categorical logic, topoi
18D30 Fibered categories
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References:
[1] Des catégories fibrées
[2] Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences (Paris), Séries A et B 281 pp A897– (1975)
[3] Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences (Paris), Séries A et B 281 pp A831– (1975)
[4] I, Séminaire Bourbaki 1959/60 (1969)
[5] Théorie des topos et cohomologie étale des schémas, Séminaire de Géométrie Algébrique du Bois-Marie 1963/64 (SGA 4) 269 (1972)
[6] Fibrations et extensions de Kan (1975)
[7] Fibrations géométiriques et théorème de Giraud (1982)
[8] Topos theory (1977)
[9] Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique du Bois-Marie 1960/61 (SGA 1) 224 pp 145– (1971)
[10] Cohomologie non abélienne (1971)
[11] Indexed categories and their applications 661 (1978)
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