Tripos theory

Introduction. One of the most important constructions in topos theory ia that of the category Shv (̂ 4) 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. To motivate what we are about to do, let us examine the concept of locale in some detail. To take care of the propositional logic of 'A -valued sets', we require first of all that A should be a Heyting algebra: that is, a partially ordered set which (considered as a category) is finitely complete and cocomplete (i.e. has all finite meets and joins) and is cartesian closed (i.e. has an implication operator -> such that a < (b->c) if and only if (a A b) < c). To handle predicate logic, we require A to be complete (equivalently, cocomplete); normally, we express this by saying that we have join and meet maps