Mathematical Structures in Computer Science

Special Issue: Realizability

Tripos theory in retrospect

a1 Cambridge University Computer Laboratory, William Gates Building, J. J. Thomson Avenue, Cambridge, CB3 0FD, U.K.


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.

(Received December 7 1999)
(Revised August 7 2000)