Tripos theory in retrospect
Andrew M. Pitts · Mathematical Structures in Computer Science · 2002
The notion of tripos (Hyland et al . 1980; Pitts 1981) was motivated by the desire to explain in what sense Higg's description of sheaf toposes as H -valued sets and Hyland's realizability toposes are instances of the same construction. The construction itself can be seen as the universal solution to the problem of realizing the predicates of a first order hyperdoctrine as subobjects in a logos with effective equivalence relations. In this note it is shown that the resulting logos is actually a topos if and only if the original hyperdoctrine satisfies a certain comprehension property. Triposes satisfy this property, but there are examples of non-triposes satisfying this form of comprehension.