Abstract

In this talk, I will give a characterisation of realisability toposes over partial combinatory algebras. More precisely, we do not characterise the bare toposes, but the toposes together with the associated ‘constant objects functor’ Delta: Set → RT(A) for a partial combinatory algebra A. By Moens’ theorem, such a functor is equivalent to a fibering of RT(A) over Set (given via glueing).

Our approach is to view realisability toposes as generalised presheaf toposes, where the underlying category is replaced by an underlying fibration. As in the non-fibred case, these generalised presheaf toposes can be characterised in terms of indecomposable projectives. To obtain a characterisation of realisability over a partial combinatory algebra, it remains to characterise the fibrations that are induced by pcas. This is achieved using techniques introduced by Hofstra and Longley in their work on categories of ‘combinatory objects’.