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Locally cartesian closed categories and type theory

Published online by Cambridge University Press:  24 October 2008

R. A. G. Seely
R. A. G. Seely
Affiliation:
John Abbott College, Ste. Anne de Bellevue, Quebec H9X 3L9, Canada
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Extract

It is well known that for much of the mathematics of topos theory, it is in fact sufficient to use a category C whose slice categories C/A are cartesian closed. In such a category, the notion of a ‘generalized set’, for example an ‘A-indexed set’, is represented by a morphism B → A of C, i.e. by an object of C/A. The point about such a category C is that C is a C-indexed category, and more, is a hyper-doctrine, so that it has a full first order logic associated with it. This logic has some peculiar aspects. For instance, the types are the objects of C and the terms are the morphisms of C. For a given type A, the predicates with a free variable of type A are morphisms into A, and ‘proofs’ are morphisms over A. We see here a certain ‘ambiguity’ between the notions of type, predicate, and term, of object and proof: a term of type A is a morphism into A, which is a predicate over A; a morphism 1 → A can be viewed either as an object of type A or as a proof of the proposition A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 95 , Issue 1 , January 1984 , pp. 33 - 48
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

REFERENCES

[1] Cartmell, J. W.. Generalized algebraic theories and contextual categories, Ph.D. Thesis, University of Oxford, 1978.Google Scholar
[2] Diller, J.. Modified realisation and the formulae-as-types notion. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, ed. Seldin, J. P. and Hindley, J. R. (Academic Press, 1980), 491–501.Google Scholar
[3] Freyd, P.. Aspects of topoi, Bull. Australian Math. Soc. 7 (1972), 1–76.CrossRefGoogle Scholar
[4] MacLane, S.. Categories for the Working Mathematician (Springer-Verlag, 1971).Google Scholar
[5] Martin-Löf, P.. An intuitionistic theory of types: predicative part. In Logic Colloquium '73, ed. Rose, H. E. and Shepherdson, J. C. (North-Holland, 1974), 73–118.Google Scholar
[6] Paré, R. and Schumacher, D.. Abstract families and the adjoint functor theorems. In Indexed Categories and Their Applications, ed. Johnstone, P. T. and Paré, R. (Springer-Verlag, 1978).Google Scholar
[7] Prawitz, D.. Natural Deduction: a Proof-theoretical Study (Almqvist and Wiksell, 1965).Google Scholar
[8] Prawitz, D.. Ideas and results in proof theory. In Proc. of the Second Scandinavian Logic Symposium, ed. Fenstad, J. E. (North-Holland, 1971), 235–307.CrossRefGoogle Scholar
[9] Seely, R. A. G.. Hyperdoctrines and natural deduction. Ph.D. Thesis, University of Cambridge, 1977.Google Scholar
[10] Seely, R. A. G.. Hyperdoctrines, natural deduction, and the Beck condition. Zeitschrift für Math. Logik und Grundlagen der Math. (To appear, 1984.)Google Scholar
[11] Seely, R. A. G.. Locally cartesian closed categories and type theory. McGill University Mathematics Report 82–22 (Montreal, 1982).Google Scholar
[12] Seely, R. A. G.. Locally cartesian closed categories and type theory. Mathematical Reports of the Academy of Science, IV, 5, 271–275. (Canada, 1982).Google Scholar