Logic and Category Theory

  • Gonzalo E. Reyes
Part of the Synthese Library book series (SYLI, volume 149)


In spite of the title, this paper deals with one aspect only of the interconnections between logic and category theory, namely the dialectics of ‘concept’ versus ‘Variable set’ or, more precisely, the connections between model theory and topos theory. The connections between proof theory and category theory have been completely left out and the interested reader is referred to [22], where a full account of the work done in that area may be found. Furthermore, since a survey article on the connections between model theory and topos theory has recently appeared [10], I shall rather concentrate on one particular problem: to formulate rigourously the ‘principle’ that ‘in the infinitely small, every function is linear’ (Although this ‘principle’ was one of the fundamental intuitions that helped to develop calculus in the 17th Century, it did not survive the ‘Arithmetization of Analysis’ in the 19th and was swept out as hopelessly wrong along with infinitesimals, fluxions, etc.).


Topological Space Category Theory Intuitionistic Logic Adjointness Condition Intended Interpretation 
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  1. [1]
    Gray, J. W., ‘Fragments of the History of Sheaf Theory’, in Applications of Sheaves (Proceedings, Durham 1977), Lecture Notes in Mathematics No. 753, Springer-Verlag 1979, pp. 1–79.Google Scholar
  2. [2]
    Higgs, D., ‘A Category Approach to Boolean Valued Set Theory’, preprint, 1973.Google Scholar
  3. [3]
    Joyal, A., ‘Les théorèmes de Chevalley-Tarski et remarques sur l’Algebre constructive’, Cahiersde Topo et Géom. Dif XVI (3) (1975), 256–258.Google Scholar
  4. [4]
    Johnstone, P. T., Topos Theory, Academic Press, 1977.Google Scholar
  5. [5]
    Kline, M., Mathematical Thought from Ancient to Modern Times, Oxford Univ. Press, 1972.Google Scholar
  6. [6]
    Kock, A., ‘A Simple Axiomatics for Differentiation’, Math. Scand. 40 (1977), 183–193.Google Scholar
  7. [7]
    Kock, A., ‘Taylor Series Calculus for Ring Objects of Line Type’, J. Pure Appl. Algebra 12 (1978), 271–293.CrossRefGoogle Scholar
  8. [8]
    Kock, A., Proprietá del’ anello generic, Notes by Anna Barbara Veit Riccioli, Aarhus, 1977.Google Scholar
  9. [9]
    Kock, A., ‘Universal Projective Geometry Via Topos Theory’, J. Pure Appl. Algebra 9 (1976), 1–24.CrossRefGoogle Scholar
  10. [10]
    Kock, A. and Reyes, G. E., ‘Doctrines in Categorical Logic’, in Handbook of Mathematical Logic, North-Holland, Amsterdam 1977, pp. 283–313.Google Scholar
  11. [11]
    Kock, A. and Reyes, G. E., ‘Manifolds in Formal Differential Geometry’, in Applications of Sheaves (Proceedings, Durham 1977), Lecture Notes in Mathematics No. 753, Springer-Verlag 1979, pp. 514–533.Google Scholar
  12. [12]
    Kock, A. and Reyes G. E., ‘Connections in Formal Differential Geometry’, in Topos Theoretic Methods in Geometry, Aarhus Various Publications Series No. 30, 1979, pp. 158–195Google Scholar
  13. [13]
    Lawvere, F. W., ‘An Elementary Theory of the Category of Sets’, Proc. Nat. Acad. Sci. U.S.A. 52 (1964), 1506–1511.CrossRefGoogle Scholar
  14. [14]
    MacLane, S., Categories for the Working Mathematician, Springer-Verlag, Heidel-berg, 1971.Google Scholar
  15. [15]
    Makkai, M. and Reyes, G. E., First Order Categorical Logic, Lecture Notes in Mathematics, No. 611, Springer-Verlag, Heidelberg, 1977.Google Scholar
  16. [16]
    Mulvey, C. J., ‘Intuitionistic Algebra and Representation of Rings’, in Recent Advances in the Representation Theory of Rings and C*-Algebras by Continuous Sections, Mem. Am. Math. Soc., No. 148, 1974.Google Scholar
  17. [17]
    Reyes, G. E., ‘Theorie des modeles et fais ceaux’, Advances in Math. 30, No. 2 (1978), 156–170.CrossRefGoogle Scholar
  18. [18]
    Reyes, G. E., ‘Cramer’s Rule in The Zariski Topos’, in Applications of Sheaves (Proceedings, Durham 1977), Lecture Notes in Mathematics No. 753, Springer-Verlag 1979, pp. 586–594.Google Scholar
  19. [19]
    Reyes, G. E. and Wraith, G. C.,4A Note on Tangent Bundles in a Category, with a Ring Object’, Math. Scand. 42 (1978), 53–63.Google Scholar
  20. [20]
    Robinson, A., Non-Standard Analysis, North-Holland, Amsterdam, 1966.Google Scholar
  21. [21]
    Shafarevich, I. R., Basic Algebraic Geometry, Springer-Verlag, Heidelberg, 1974.Google Scholar
  22. [22]
    Szabo, M. E., Algebra of Proofs, North-Holland, Amsterdam, 1977.Google Scholar
  23. [23]
    Veit-Riccioli, A. B., ‘Il forcing come principio logico per la construzione dei fasci’, preprint, 1977.Google Scholar
  24. [24]
    Wraith, G. C., ‘Lectures on Elementary Topoi’, in Model Theory and Topoi, Lecture Notes in Mathematics No. 445, Springer-Verlag, Heidelberg, 1975, pp. 114–206.Google Scholar
  25. Added in Proof. Since the writing up of this paper, E. Dubric has built models for Kock’s axiom as well as other axioms introduced in [11], which comprise ‘classical’ differential manifolds. For these developments see the book quoted in [12] as well asGoogle Scholar
  26. [25]
    Reyes, G. E. (Editor), Géometrie Différentielle Synthétique, Fasciarles 1, 2, Rapport de Recherches du DMS Nos. 80–11, 80–12, Université de Montréal, 1980.Google Scholar

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© D. Reidel Publishing Company 1981

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  • Gonzalo E. Reyes

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