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The limit–colimit coincidence theorem for -categories

Published online by Cambridge University Press:  25 March 2010

MATEUSZ KOSTANEK  and
PAWEŁ WASZKIEWICZ
MATEUSZ KOSTANEK
Affiliation:
Theoretical Computer Science, Jagiellonian University, ul. S. Łojasiewicza 6, 30-348 Kraków, Poland Email: kostanek@gmail.com; pqw@tcs.uj.edu.pl
PAWEŁ WASZKIEWICZ
Affiliation:
Theoretical Computer Science, Jagiellonian University, ul. S. Łojasiewicza 6, 30-348 Kraków, Poland Email: kostanek@gmail.com; pqw@tcs.uj.edu.pl
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Abstract

We prove that the category of -cocomplete separated -categories has bilimits of expanding sequences. This result generalises on various levels the well-known theorem of domain theory that guarantees the existence of bilimits in the category of directed-complete posets and Scott-continuous maps.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Special Issue 2: Domains , April 2010 , pp. 267 - 284
Copyright
Copyright © Cambridge University Press 2010

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References

Abramsky, S. and Jung, A. (1994) Domain Theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science 3, Oxford University Press 1–168.Google Scholar
Albert, M. H. and Kelly, G. M. (1998) The closure of a class of colimits. Journal of Pure and Applied Algebra 51 1–17.CrossRefGoogle Scholar
America, P. and Rutten, J. J. M. M (1989) Solving Reflexive Domain Equations in a Category of Complete Metric Spaces. Journal of Computer and System Sciences 39 (3)343–375.CrossRefGoogle Scholar
Barr, M. (1970) Relational Algebras. In: Reports of the Midwest Category Seminar, IV. Springer-Verlag Lecture Notes in Computer Science 137 39–55.Google Scholar
Bonsangue, M. M., van Breugel, F. and Rutten, J. J. M. M (1998) Generalized Metric Spaces: Completion, Topology, and Powerdomains via the Yoneda Embedding. Theoretical Computer Science 193 (1–2)1–51.CrossRefGoogle Scholar
Clementino, M. M. and Hofmann, D. (2003) Topological Features of Lax Algebras. Applied Categorical Structures 11 267–286.CrossRefGoogle Scholar
Clementino, M. M. and Tholen, W. (2003) Metric, Topology and Multicategory – a Common Approach. Journal of Pure and Applied Algebra 179 13–47.CrossRefGoogle Scholar
Clementino, M. M., Hofmann, D. and Tholen, W. (2004) One Setting for All: Metric, Topology, Uniformity, Approach Structure. Applied Categorical Structures 12 (2)127–154.CrossRefGoogle Scholar
Flagg, R. C. (1997) Quantales and Continuity Spaces. Algebra Universalis 37 257–276.CrossRefGoogle Scholar
Flagg, R. C. and Kopperman, R. (1995) Fixed Points and Reflexive Domain Equations in Categories of Continuity Spaces. Electronic Notes in Theoretical Computer Science 1.CrossRefGoogle Scholar
Flagg, R. C. and Kopperman, R. (1997) Continuity Spaces: Reconciling Domains and Metric Spaces. Theoretical Computer Science 177 (1)111–138.CrossRefGoogle Scholar
Flagg, R. C. and Sünderhauf, P. (2002) The Essence of Ideal Completion in Quantitative Form. Theoretical Computer Science 278 (1–2)141–158.CrossRefGoogle Scholar
Flagg, R.C., Sünderhauf, P. and Wagner, K.R. (1996) A Logical Approach to Quantitative Domain Theory. Topology Atlas Preprint 23 (available on-line at: http://at.yorku.ca/e/a/p/p/23.htm.)Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous Lattices and Domains. Encyclopedia of mathematics and its applications 93, Cambridge University Press.Google Scholar
Heckmann, R. (2007) Similarity, Topology, and Uniformity. Presented at Dagstuhl workshop ‘Logic, Computability and Topology in Computer Science: A New Perspective for Old Disciplines’ (submitted for publication). (Available on-line at: http://rw4.cs.uni-sb.de/~heckmann/domains.html.)Google Scholar
Hofmann, D. (2007) Topological Theories and Closed Objects. Advances in Mathematics 215 789–824.CrossRefGoogle Scholar
Kelly, G. M. (1982) Basic Concepts of Enriched Category Theory. London Mathematical Society Lecture Notes 64.Google Scholar
Kelly, G. M. and Lack, S. (2000) On the Monadicity of Categories with Chosen Colimits. Theory and Applications of Categories 7 148–170.Google Scholar
Kelly, G. M. and Schmitt, V. (2005) Notes on Enriched Categories with Colimits of Some Class. Theory and Applications of Categories 14 399–423.Google Scholar
Kock, A. (1995) Monads for which Structures are Adjoint to Units. Journal of Pure and Applied Algebra 104 41–59.CrossRefGoogle Scholar
Lai, H. and Zhang, D. (2007) Complete and Directed-complete Ω-categories. Theoretical Computer Science 388 1–25.CrossRefGoogle Scholar
Lawvere, F. W. (2002) Metric Spaces, Generalized Logic and Closed Categories. Reprints in Theory and Applications of Categories 1 1–37.Google Scholar
Manes, E. G. (1974) Compact Hausdorff Objects. General Topology and Applications 4 341–360.CrossRefGoogle Scholar
Rosenthal, K. (1990) Quantales and Their Applications, Pittman Research Notes in Mathematics series 234, Longman.Google Scholar
Rutten, J. J. M. M. (1996) Elements of Generalized Ultrametric Domain Theory. Theoretical Computer Science 170 349–381.CrossRefGoogle Scholar
Schmitt, V. (2006) Flatness, Preorders and General Metric Spaces. To appear in Georgian Mathematical Journal. (Available at arXiv:math.CT/0602463.)Google Scholar
Vickers, S. (2005) Localic Completion of Generalized Metric Spaces I. Theory and Applications of Categories 14 328–356.Google Scholar
Wagner, K. R. (1994) Solving Recursive Domain Equations with Enriched Categories, Ph.D. Thesis, Carnegie Mellon University.Google Scholar
Waszkiewicz, P. (2009) On Domain Theory over Girard Quantales. Fundamenta Informaticae 92 1–24.CrossRefGoogle Scholar
Zhang, Q.-Y. and Fan, L. (2005) Continuity in Quantitative Domains. Fuzzy Sets and Systems 154 (1)118–131.CrossRefGoogle Scholar