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Fibrations with indeterminates: contextual and functional completeness for polymorphic lambda calculi

Published online by Cambridge University Press:  04 March 2009

Claudio Hermida  and
Bart Jacobs
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Abstract

Lambek used categories with indeterminates to capture explicit variables in simply typed λ-calculus. He observed that such categories with indeterminates can be described as Kleisli categories for suitable comonads. They account for ‘functional completeness’ for Cartesian (closed) categories.

Here we refine this analysis, by distinguishing ‘contextual’ and ‘functional’ completeness, and extend it to polymorphic λ-calculi. Since the latter are described as certain fibrations, we are lead to consider indeterminates, not only for ordinary categories, but also for fibred categories. Following a 2-categorical generalisation of Lambek's approach, such fibrations with indeterminates are presented as 'simple slices' in suitable 2-categories of fibrations; more precisely, as Kleisli objects. It allows us to establish contextual and functional completeness results for some polymorphic calculi.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 5 , Issue 4 , December 1995 , pp. 501 - 531
Copyright
Copyright © Cambridge University Press 1995

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