The Cambridge Natural Sciences Tripos

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 pred...

Foucault and the Natural Sciences

in Humanities , Social Sciences , and Natural Sciences

Tripos Theory in Retrospect

The notion of tripos (Hyland, Johnstone, and Pitts 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 subobjec...

University of Natural Sciences

We start with irrationality proofs. Historically, the first ones concerned irrational algebraic numbers, like the square roots of non square positive integers. Next, the theory of continued fraction expansion provided a very useful tool. Among the first proofs of irrationality for numbers which are now known to be transcendental are the ones by H. Lambert and L. Euler, in the XVIIIth century, f...