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Domains occur among spaces as strict algebras among lax

Published online by Cambridge University Press:  30 March 2011

RORY B. B. LUCYSHYN-WRIGHT
RORY B. B. LUCYSHYN-WRIGHT*
Affiliation:
York University, 4700 Keele St., Toronto, ON, CanadaM3J 1P3 Email: rorylw@mathstat.yorku.ca
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Abstract

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Whereas Alan Day showed that the continuous lattices are the algebras of a filter monad on Set, we employ the theory of lax algebras (as developed by Barr, Pisani, Clementino, Hofmann, Tholen, Seal and others) to broaden this characterisation to a description of the wider class of continuous dcpos as algebras of a lax filter monad. Building on an axiomatisation of topological spaces through convergence as lax algebras of a lax extension of the filter monad to a category of relations, we show that those topological spaces whose associated lax algebra is in fact a strict algebra are what M. Erné called the C-spaces. The sober C-spaces are precisely the continuous dcpos under the Scott topology, and we discuss how the possibly little-known C-spaces, which have been studied by B. Banaschewski, J. D. Lawson, R.-E. Hoffmann, M. Erné and G. Wilke, very directly capture an essential topological notion of approximation inherent in the continuous dcpos, and hence provide a natural topological concept of domain.

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Type
Paper
Information
Mathematical Structures in Computer Science , Volume 21 , Issue 3 , June 2011, pp. 647 - 670
Copyright
Copyright © Cambridge University Press 2011

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