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Gabriel–Ulmer duality and Lawvere theories enriched over a general base

Part of: JFP Research Articles

Published online by Cambridge University Press:  01 July 2009

STEPHEN LACK  and
JOHN POWER
STEPHEN LACK
Affiliation:
School of Computing and Mathematics, University of Western Sydney, Locked Bag 1797, Penrith South DC, NSW 1797, Australia (e-mail: s.lack@uws.edu.au)
JOHN POWER
Affiliation:
Department of Computer Science, University of Bath, Claverton Down, Bath BA2 7AY, UK (e-mail: a.j.power@bath.ac.uk)
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Abstract

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Motivated by the search for a body of mathematical theory to support the semantics of computational effects, we first recall the relationship between Lawvere theories and monads on Set. We generalise that relationship from Set to an arbitrary locally presentable category such as Poset and ωCpo or functor categories such as [Inj, Set] and [Inj, ωCpo]. That involves allowing the arities of Lawvere theories to be extended to being size-restricted objects of the locally presentable category. We develop a body of theory at this level of generality, in particular explaining how the relationship between generalised Lawvere theories and monads extends Gabriel–Ulmer duality.

Type
Articles
Information
Journal of Functional Programming , Volume 19 , Issue 3-4 , July 2009 , pp. 265 - 286
Copyright
Copyright © Cambridge University Press 2009

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