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Tripos theory

Published online by Cambridge University Press:  24 October 2008

J. M. E. Hyland ,
P. T. Johnstone  and
A. M. Pitts
J. M. E. Hyland
Affiliation:
Department of Pure Mathematics, University of Cambridge, England
P. T. Johnstone
Affiliation:
Department of Pure Mathematics, University of Cambridge, England
A. M. Pitts
Affiliation:
Department of Pure Mathematics, University of Cambridge, England
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Extract

One of the most important constructions in topos theory ia that of the category Shv (A) of sheaves on a locale (= complete Heyting algebra) A. Normally, the objects of this category are described as ‘presheaves on A satisfying a gluing condition’; but, as Higgs(7) and Fourman and Scott(5) have observed, they may also be regarded as ‘sets structured with an A-valued equality predicate’ (briefly, ‘A-valued sets’). From the latter point of view, it is an inessential feature of the situation that every sheaf has a canonical representation as a ‘complete’ A-valued set. In this paper, our aim is to investigate those properties which A must have for us to be able to construct a topos of A-valued sets: we shall see that there is one important respect, concerning the relationship between the finitary (propositional) structure and the infinitary (quantifier) structure, in which the usual definition of a locale may be relaxed, and we shall give a number of examples (some of which will be explored more fully in a later paper (8)) to show that this relaxation is potentially useful.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 88 , Issue 2 , September 1980 , pp. 205 - 232
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

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