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.
Research supported by SERC grant RR30735.
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References
R. Brown. Topology: a geometric account of general topology, homotopy types and the fundamental grupoid. Halsted Press, 1988.
A. Carboni, M.C. Pedicchio, and G. Rosolini, editors. Category Theory, volume 1488 of Lecture Notes in Mathematics. Springer-Verlag, 1991.
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.
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).
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.
M. P. Fiore. Order-enrichment for categories of partial maps. Mathematical Structures in Computer Science, 1994. To appear.
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.
J.M.E. Hyland. First steps in synthetic domain theory. In [CPR91], pages 95–104, 1991.
A. Kock. Monads on symmetric monoidal closed categories. Arch. Math. (Basel), pages 1–10, 1970.
A. Kock. Algebras for the partial map classifier monad. In [CPR91], pages 262–278, 1991.
A. Kock. Monads for which structures are adjoints to units. Manuscript (to appear in the Journal of Pure and Applied Algebra), 1994.
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.
W. Phoa. Domain Theory in Realizability Toposes. PhD thesis, University of Cambridge, 1990. (Also CST-82-91, University of Edinburgh).
G. Rosolini. Continuity and Effectiveness in Topoi. PhD thesis, University of Oxford, 1986.
R. Street. Categorical structures. Manuscript (to appear in the Handbook of Algebra volume 2, Elsevier, North Holland), November 1992.
P. Taylor. The fixed point property in synthetic domain theory. In 6 th LICS Conference, pages 152–160. IEEE, 1991.
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Fiore, M.P. (1995). Lifting as a KZ-doctrine. In: Pitt, D., Rydeheard, D.E., Johnstone, P. (eds) Category Theory and Computer Science. CTCS 1995. Lecture Notes in Computer Science, vol 953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60164-3_24
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DOI: https://doi.org/10.1007/3-540-60164-3_24
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