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The theory of semi-functors

Published online by Cambridge University Press:  04 March 2009

Raymond Hoofman
Raymond Hoofman
Affiliation:
Department of Mathematics and Computer Science, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands
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Abstract

The notion of semi-functor was introduced in Hayashi (1985) in order to make possible a category-theoretical characterization of models of the non-extensional typed lambda calculus. Motivated by the further use of semi-functors in Martini (1987), Jacobs (1991) and Hoofman (1992a), (1992b) and (1992c), we consider the general theory of semi-functors in this paper. It turns out that the notion of semi natural transformation plays an important part in this theory, and that various categorical notions involving semi-functors can be viewed as 2-categorical notions in the 2-category of categories, semi-functors and semi natural transformations. In particular, we find that the notion of normal semi-adjunction as defined in Hayashi (1985) is the canonical generalization of the notion of adjunction to the world of semi-functors. Further topics covered in this paper are the relation between semi-functors and splittings, the Karoubi envelope construction, semi-comonads, and a semi-adjoint functor theorem.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 3 , Issue 1 , March 1993 , pp. 93 - 128
Copyright
Copyright © Cambridge University Press 1993

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References

Berry, G. (1987) Stable Models of Typed λ-Calculi. In: Ausiello, G. and Böhm, C. (eds.), Automata, Languages and Programming, Fifth Colloquium. Springer-Verlag Lect. Notes in Comp. Sci. 62 72–89.CrossRefGoogle Scholar
Freyd, P. A. and Scedrov, A. (1989) Categories, Allegories, North-Holland, Amsterdam.Google Scholar
Girard, J.-Y. (1986) The System F of Variable Types, Fifteen Years Later. Theor. Comput. Sci. 45 159–192.CrossRefGoogle Scholar
Girard, J.-Y. (1987) Linear Logic. Theor. Comput. Sci. 50 1–102.CrossRefGoogle Scholar
Hayashi, S. (1985) Adjunction of Semifunctors: Categorical Structures in Non-Extensional Lambda-Calculus. Theor. Comput. Sci. 41 95–104.CrossRefGoogle Scholar
Hoofman, R. (1992a) Continuous Information Systems. Information and Computation (to appear).Google Scholar
Hoofman, R. (1992b) Non-Stable Models of Linear Logic. Logical Foundations of Computer Science, Proceedings. Springer-Verlag Lect. Notes in Comp. Sci. (to appear).Google Scholar
Hoofman, R. (1992c) Non-Stable Models of Linear Logic, Ph.D. thesis, Utrecht University, Utrecht.Google Scholar
Hoofman, R., AND Schellinx, H. (1991) Collapsing Graph Models by Preorders. In: Pitt, D. H., Curien, P.-L., Abramsky, S., Pitts, A. M., Poigné, A. and Rydeheard, D. E. (eds.), Category Theory and Computer Science, Proceedings. Springer-Verlag Lect. Notes in Comp. Sci. 530 53–73.CrossRefGoogle Scholar
Jacobs, B. (1991) Semantics of Second Order Lambda Calculus. Mathematical Structures in Computer Science 1 (3).CrossRefGoogle Scholar
Karoubi, M. (1978) K-theory, An Introduction, Springer, Berlin/New-York.CrossRefGoogle Scholar
Koymans, C. P. J. (1982) Models of the Lambda Calculus. Inform. and Control 52 306–322.CrossRefGoogle Scholar
Kelly, G. M. and Street, R. (1974) Review of the Elements of 2-Categories. In: Category Seminar.Springer-Verlag Lect. Notes in Mathematics 420 75–103.CrossRefGoogle Scholar
Lambek, J. and Scott, P. J. (1986) Introduction to Higher Order Categorical Logic, Studies in Advanced Mathematics 7, Cambridge University Press.Google Scholar
Mac Lane, S. (1971) Categories for the Working Mathematician, Springer-Verlag, New-York.CrossRefGoogle Scholar
Martini, S. (1987) An Interval Model for Second Order Lambda Calculus. In: Pitt, D. H., Poigné, A. and Rydeheard, D. (eds.), Category Theory and Computer Science, Proceedings. Springer-Verlag Lect. Notes in Comp. Sci. 283 219–237.Google Scholar
Matsumoto, M. (1989) On the Yoneda Lemma and Adjunctions, Master’s Thesis, Dept. of Inform. Sci., Faculty of Sci., University of Tokyo.Google Scholar
Obtułowicz, A. and Wiweger, A. (1982) Categorical, Functional and Algebraic Aspects of the Type-Free Lambda Calculus. Universal Algebra and Applications, Banach Center Publications 9 399–422.Google Scholar
Wiweger, A. (1984) Pre-adjunction and,λ-algebraic theories. Colloq. Math. 48 (2) 153–165.CrossRefGoogle Scholar