Tripos theory

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.