Hostname: page-component-84b7d79bbc-rnpqb Total loading time: 0 Render date: 2024-07-31T14:36:12.891Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. (1991) Domain theory in logical form. Ann. Pure Appl. Logic 51 1–77.CrossRefGoogle Scholar
Abramsky, S. and Jung, A. (1994) Domain theory. In: Handbook of logic in computer science 3Oxford University Press 1–168.Google Scholar
Adámek, J., Herrlich, H. and Strecker, G. E. (1990) Abstract and concrete categories, John Wiley & Sons Inc.Google Scholar
Banaschewski, B. (1977) Essential extensions of T 0-spaces. General Topology and Appl. 7 233–246.CrossRefGoogle Scholar
Barr, M. (1970) Relational algebras. Springer-Verlag Lecture Notes in Mathematics 137 39–55.CrossRefGoogle Scholar
Borceux, F. (1994) Handbook of categorical algebra 1, Cambridge University Press.Google Scholar
Clementino, M. M. and Hofmann, D. (2003) Topological features of lax algebras. Appl. Categ. Structures 11 267–286.CrossRefGoogle Scholar
Clementino, M. M., Hofmann, D. and Tholen, W. (2004) One setting for all: metric, topology, uniformity, approach structure. Appl. Categ. Structures 12 127–154.CrossRefGoogle Scholar
Clementino, M. M. and Tholen, W. (2003) Metric, topology and multicategory – a common approach. J. Pure Appl. Algebra 179 13–47.Google Scholar
Day, A. (1975) Filter monads, continuous lattices and closure systems. Canad. J. Math. 27 50–59.Google Scholar
Erné, M. (1980) Vollständig distributive topologien und idempotente relationen. Abstracts Deutsche Mathematiker-Vereinigung, Jahrestagung Dortmund.Google Scholar
Erné, M. (1981) Scott convergence and Scott topology in partially ordered sets, ii. Springer-Verlag Lecture Notes in Mathematics 871 61–96.CrossRefGoogle Scholar
Erné, M. (1991) The ABC of order and topology. In: Category theory at work (Bremen, 1990), Heldermann 57–83.Google Scholar
Erné, M. (1999) -continuous posets and their topological manifestation. Appl. Categ. Structures 7 31–70.CrossRefGoogle Scholar
Erné, M. and Wilke, G. (1983) Standard completions for quasiordered sets. Semigroup Forum 27 351–376.Google Scholar
Escardó, M. H. (1997) Injective spaces via the filter monad. Topology Proc. 22 97–110.Google Scholar
Escardó, M. H. (1998) Properly injective spaces and function spaces. Topology Appl. 89 75–120.Google Scholar
Escardó, M. H. (2004) Synthetic topology of data types and classical spaces. Electronic Notes in Theoretical Computer Science 87 21–156.Google Scholar
Escardó, M. H. and Flagg, R. C. (1999) Semantic domains, injective spaces and monads (extended abstract). Electronic Notes in Theoretical Computer Science 20.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous lattices and domains, Cambridge University Press.CrossRefGoogle Scholar
Hoffmann, R.-E. (1979) Projective sober spaces. In: Structure of topological categories (Proc. Conf., Univ. Bremen, Bremen, 1978) 18 109–153.Google Scholar
Hoffmann, R.-E. (1981) Continuous posets, prime spectra of completely distributive complete lattices, and hausdorff compactifications. Springer-Verlag Lecture Notes in Mathematics 871 159–208.Google Scholar
Hofmann, D. and Tholen, W. (2006) Kleisli compositions for topological spaces. Topology Appl. 153 2952–2961.CrossRefGoogle Scholar
Johnstone, P. T. (1982) Stone spaces, Cambridge University Press.Google Scholar
Lawson, J. (1997) The round ideal completion via sobrification. Topology Proc. 22 261–274.Google Scholar
Lawson, J. D. (1979) The duality of continuous posets. Houston J. Math. 5 357–386.Google Scholar
Lawvere, F. W. (1963) Functorial semantics of algebraic theories, Ph.D. thesis, Columbia University, New York. (Also available in: Repr. Theory Appl. Categ. (2004) 5 1–121.)Google Scholar
Lawvere, F. W. (1973) Metric spaces, generalized logic, and closed categories. Rend. Sem. Mat. Fis. Milano 43. (Also available in: Repr. Theory Appl. Categ. (2002) 1 1–37.)Google Scholar
Linton, F. E. J. (1969) An outline of functorial semantics. In: Sem. on Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer-Verlag 7–52. (Also available in: Repr. Theory Appl. Categ. (2008) 18 1–303.)Google Scholar
MacLane, S. Lane, S. (1998) Categories for the working mathematician, second edition, Springer-Verlag.Google Scholar
Manes, E. (1969) A triple theoretic construction of compact algebras. Springer-Verlag Lecture Notes in Mathematics 80 91–118.CrossRefGoogle Scholar
Manes, E. G. (1976) Algebraic theories, Springer-Verlag.CrossRefGoogle Scholar
Mislove, M. W. (1998) Topology, domain theory and theoretical computer science. Topology Appl. 89 3–59.Google Scholar
Pisani, C. (1999) Convergence in exponentiable spaces. Theory Appl. Categ. 5 148–162.Google Scholar
Schalk, A. (1993a) Algebras for generalized power constructions, Ph.D. thesis, Technische Hochschule Darmstadt.Google Scholar
Schalk, A. (1993b) Domains arising as algebras for powerspace constructions. J. Pure Appl. Algebra 89 305–328.CrossRefGoogle Scholar
Scott, D. (1972) Continuous lattices. Springer-Verlag Lecture Notes in Mathematics 274 97–136.Google Scholar
Seal, G. J. (2005) Canonical and op-canonical lax algebras. Theory Appl. Categ. 14 221–243.Google Scholar
Seal, G. J. (2010) Order-adjoint monads and injective objects. J. Pure Appl. Algebra 214 778–796.Google Scholar
Smyth, M. B. (1983) Power domains and predicate transformers: a topological view. Springer-Verlag Lecture Notes in Computer Science 154 662–675.CrossRefGoogle Scholar
Smyth, M. B. (1992) Topology. In: Handbook of logic in computer science 1, Oxford University Press 641–761.Google Scholar
Street, R. (1974) Fibrations and Yoneda's lemma in a 2-category. Springer-Verlag Lecture Notes in Mathematics 420 104–133.CrossRefGoogle Scholar
Taylor, P. (1990) An algebraic approach to stable domains. J. Pure Appl. Algebra 64 171–203.Google Scholar
Tholen, W. (2007) Lax-algebraic methods in general topology. Lecture notes of the Summer School in Categorical Methods in Algebra and Topology, Haute Bodeux (Belgium) 2007. (Available at www.math.yorku.ca/~tholen/.)Google Scholar
Tholen, W. (2009) Ordered topological structures. Topology Appl. 156 2148–2157.Google Scholar
Vickers, S. (1989) Topology via logic, Cambridge University Press.Google Scholar
Wyler, O. (1981) Algebraic theories of continuous lattices. Springer-Verlag Lecture Notes in Mathematics 871 390–413.Google Scholar
Wyler, O. (1985) Algebraic theories for continuous semilattices. Arch. Ration. Mech. Anal. 90 99–113.CrossRefGoogle Scholar