Advertisement

Lifting as a KZ-doctrine

  • Marcelo P. Fiore
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 953)

Abstract

In a cartesian closed category with an initial object and a dominance that classifies it, an intensional notion of approximation between maps —the path relation (c.f. link relation)— is defined. It is shown that if such a category admits strict/upper-closed factorisations then it preorderenriches (as a cartesian closed category) with respect to the path relation. By imposing further axioms we can, on the one hand, endow maps and proofs of their approximations (viz. paths) with the 2-dimensional algebraic structure of a sesqui-category and, on the other, characterise lifting as a preorder-enriched lax colimit. As a consequence of the latter the lifting (or partial map classifier) monad becomes a KZ-doctrine.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bro88]
    R. Brown. Topology: a geometric account of general topology, homotopy types and the fundamental grupoid. Halsted Press, 1988.Google Scholar
  2. [CPR91]
    A. Carboni, M.C. Pedicchio, and G. Rosolini, editors. Category Theory, volume 1488 of Lecture Notes in Mathematics. Springer-Verlag, 1991.Google Scholar
  3. [Fio93]
    M. P. Fiore. Cpo-categories of partial maps. In CTCS-5 (Category Theory and Computer Science Fifth Biennial Meeting), pages 45–49. CWI, September 1993.Google Scholar
  4. [Fio94a]
    M. P. Fiore. Axiomatic Domain Theory in Categories of Partial Maps. PhD thesis, University of Edinburgh, 1994. (Available as technical report ECS-LFCS-94-307 or from http://www.dcs.ed.ac.uk/home/mf/thesis.dvi.Z).Google Scholar
  5. [Fio94b]
    M. P. Fiore. First steps on the representation of domains. Manuscript (available from http://www.dcs.ed.ac.uk/home/mf/path.dvi), December 1994.Google Scholar
  6. [Fio94c]
    M. P. Fiore. Order-enrichment for categories of partial maps. Mathematical Structures in Computer Science, 1994. To appear.Google Scholar
  7. [FP94]
    M.P. Fiore and G.D. Plotkin. An axiomatisation of computationally adequate domain theoretic models of FPC. In 9 th LICS Conference, pages 92–102. IEEE, 1994.Google Scholar
  8. [Hyl91]
    J.M.E. Hyland. First steps in synthetic domain theory. In [CPR91], pages 95–104, 1991.Google Scholar
  9. [Koc70]
    A. Kock. Monads on symmetric monoidal closed categories. Arch. Math. (Basel), pages 1–10, 1970.Google Scholar
  10. [Koc91]
    A. Kock. Algebras for the partial map classifier monad. In [CPR91], pages 262–278, 1991.Google Scholar
  11. [Koc94]
    A. Kock. Monads for which structures are adjoints to units. Manuscript (to appear in the Journal of Pure and Applied Algebra), 1994.Google Scholar
  12. [Mog86]
    E. Moggi. Categories of partial morphisms and the partial lambda-calculus. In Proceedings Workshop on Category Theory and Computer Programming, Guildford 1985, volume 240 of Lecture Notes in Computer Science, pages 242–251. Springer-Verlag, 1986.Google Scholar
  13. [Pho90]
    W. Phoa. Domain Theory in Realizability Toposes. PhD thesis, University of Cambridge, 1990. (Also CST-82-91, University of Edinburgh).Google Scholar
  14. [Ros86]
    G. Rosolini. Continuity and Effectiveness in Topoi. PhD thesis, University of Oxford, 1986.Google Scholar
  15. [Str92]
    R. Street. Categorical structures. Manuscript (to appear in the Handbook of Algebra volume 2, Elsevier, North Holland), November 1992.Google Scholar
  16. [Tay91]
    P. Taylor. The fixed point property in synthetic domain theory. In 6 th LICS Conference, pages 152–160. IEEE, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Marcelo P. Fiore
    • 1
  1. 1.Department of Computer Science Laboratory for Foundations of Computer ScienceUniversity of EdinburghEdinburghScotland

Personalised recommendations