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Local fibred right adjoints are polynomial

Published online by Cambridge University Press:  03 October 2012

ANDERS KOCK
Affiliation:
Matematisk Institut, Aarhus Universitet, Denmark Email: [email protected]
JOACHIM KOCK
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Spain Email: [email protected]

Abstract

For any locally cartesian closed category ℰ, we prove that a local fibred right adjoint between slices of ℰ is given by a polynomial. The slices in question are taken in a well-known fibred sense.

Type
Paper
Copyright
Copyright © Cambridge University Press 2012

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References

Abbott, M., Altenkirch, T. and Ghani, N. (2003) Categories of containers. In: Foundations of software science and computation structures. Springer-Verlag Lecture Notes in Computer Science 2620 2338.CrossRefGoogle Scholar
Borceux, F. (1994) Handbook of categorical algebra. Volume 2: Categories and structures, Cambridge University Press.Google Scholar
Carboni, A. and Johnstone, P. (1995) Connected limits, familial representability and Artin glueing. Mathematical Structures in Computer Science 5 441459. (With corrigenda in Mathematical Structures in Computer Science 14 (2004) 185–187.)CrossRefGoogle Scholar
Gambino, N. and Hyland, M. (2004) Wellfounded trees and dependent polynomial functors. In: Berardi, S., Coppo, M. and Damiani, F. (eds.) TYPES 2003. Springer-Verlag Lecture Notes in Computer Science 3085 210225.CrossRefGoogle Scholar
Gambino, N. and Kock, J. (2009) Polynomial functors and polynomial monads. (Available at arXiv:0906.4931. To be published in Mathematical Proceedings of the Cambridge Philosophical Society.)Google Scholar
Johnstone, P. (1997) Cartesian monads on toposes. Journal of Pure and Applied Algebra 116 199220.CrossRefGoogle Scholar
Johnstone, P. (2002) Sketches of an elephant: a topos theory compendium: Volume 1, The Clarendon Press.Google Scholar
Kock, A. (1972) Strong functors and monoidal monads. Archiv der Mathematik 23 113120.CrossRefGoogle Scholar
Kock, A. and Reyes, G. E. (1999) Aspects of fractional exponent functors. Theory and Applications of Categories 5 251265.Google Scholar
Manes, E. G. and Arbib, M. A. (1986) Algebraic approaches to program semantics, Springer-Verlag.CrossRefGoogle Scholar
Moerdijk, I. and Palmgren, E. (2000) Wellfounded trees in categories. Annals of Pure and Applied Logic 104 189218.CrossRefGoogle Scholar
Streicher, T. (1999) Fibred categories à la Jean Bénabou. Notes, available from http://www.mathematik.tu-darmstadt.de/~streicher/.Google Scholar
Weber, M. (2007) Familial 2-functors and parametric right adjoints. Theory and Applications of Categories 18 665732.Google Scholar
Yetter, D. (1987) On right adjoints to exponential functors. Journal of Pure and Applied Algebra 45 287304. (With corrigenda in Journal of Pure and Applied Algebra 58 (1989) 103–105.)CrossRefGoogle Scholar