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The glueing construction and lax limits

Published online by Cambridge University Press:  04 March 2009

Harold Simmons
Harold Simmons
Affiliation:
Department of Computer Science, The University, Manchester M13 9PL, England
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Abstract

Having started life as a way of reconstructing a topological space from a pair of complementary subspaces, the glueing construction has found employment in a wide range of different roles, from the construction of free distributive lattices to a supporting part in the 2-categorical analysis of types theories. In this latter role the construction appears to be a fundamental factor in the behaviour of higher order proof theory. What is going on here? Before that can be answered we need at least a less ad hoc description of the construction. In this paper I set down what is, I believe, the beginnings of a coherent account of the algebraic version of glueing. As well as the abstract theory, I give a good selection of different examples to illustrate the diverse nature of the uses of the construction.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 4 , Issue 4 , December 1994 , pp. 393 - 431
Copyright
Copyright © Cambridge University Press 1994

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References

Abramsky, S. and Vickers, S. (1990) Quantales, observational logic, and process semantics, Imperial College Research Report DOC 90/1.Google Scholar
Grothendieck, A. and Verdier, J. L. (1972) Théories des Topos' (SGA 4, exposé I - VI). Second edition. Springer-Verlag Lecture Notes in Mathematics 269–270.Google Scholar
Isbell, J. R. (1982) Direct limits of meet-continuous lattices. J. Pure Appl. Algebra 23 33–35.CrossRefGoogle Scholar
Johnstone, P. T. (1977) Topos theory, Academic Press.Google Scholar
Johnstone, P. T. (1982) Stone spaces, Cambridge University Press.Google Scholar
Lambek, J. and Scott, P. J. (1986) Introduction to higher order categorical logic, Cambridge University Press.Google Scholar
Kelly, G. M. (1980) Basic concepts of enriched category theory, Cambridge University Press.Google Scholar
Macnab, D. S. (1976) An algebraic study of modal operators on heyting algebras with applications to topology and sheafification, Doctoral dissertation, University of Aberdeen.Google Scholar
Moerdijk, I. (1982) Glueing topoi and higher order disjunction and existence. In: Troelstra, A. S. and D., van Dalen (eds.) The L. E. J. Brouwer Centeniary Symposium, North-Holland.Google Scholar
Rosenthal, K. I. (1992) Girard quantaloids. Math. Struct, in Comp. Science 2 93–108.CrossRefGoogle Scholar
Sikorski, R. (1962) Applications of topology to the foundations of mathematics. In: General topology and its Relation to Modern Analysis and Algebra (Proceedings of the 1961 Prague Symposium), Academic Press 322–330.Google Scholar
Simmons, H. (1993) The glueing construction and lax limits (with applications to categories of structured posets), Tech report no. umcs-93–6–2, Department of Computer Science, University of Manchester.Google Scholar
Smyth, M. B. and Plotkin, G. D. (1983) The category-theoretic solution of recursive domain equations. SIAM J. Computing 11 761–783.CrossRefGoogle Scholar
Wand, M. (1979) Fixed-point constructions in order-enriched categories. Theoret. Comput. Sci. 8 13–30.CrossRefGoogle Scholar
Wraith, G. C. (1974) Artin glueing. J. Pure and Applied Algebra 4 345–348.CrossRefGoogle Scholar
Wright, J. B., Wagner, E. G. and Thatcher, J. W. (1978) A uniform approach to inductive posets and inductive closure. Theoret. Comput. Sci. 8 57–77.CrossRefGoogle Scholar