We show that every geometric morphism between realizability toposes satisfies the condition that its inverse image commutes with the ‘constant object’ functors, which was assumed by John Longley in his pioneering study of such morphisms. We also provide the answer to something which was stated as an open problem on Jaap van Oosten’s book on realizability toposes: if a subtopos of a realizabilit...

Plotkin [1976] introduced a powerdomain construction on domains in order to give semantics to a non-deterministic binary choice constructor, and later [1979] characterised it as the free semilattice. Smyth [1983] and Winskel [1985] showed that it could be interpreted in terms of modal predicate transformers and Robinson [1986] recognised it as a special case of Johnstone’s [1982] Vietoris const...

Blass, A. and A. Scedrov, Complete topoi representing models of set theory, Annals of Pure and Applied Logic 57 (1992) l-26. By a model of set theory we mean a Boolean-valued model of Zermelo-Fraenkel set theory allowing atoms (ZFA), which contains a copy of the ordinary universe of (two-valued, pure) sets as a transitive subclass; examples include Scott-Solovay Boolean-valued models and their ...

In the effective topos there exists a chain-complete distributive lattice with a monotone and progressive endomap which does not have a fixed point. Consequently, the Bourbaki-Witt theorem and Tarski’s fixed-point theorem for chain-complete lattices do not have constructive (topos-valid) proofs.

We show that the classifying topos for the theory of fields does not satisfy De Morgan’s law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields.

Coquand’s cubical set model for homotopy type theory provides the basis for a computational interpretation of the univalence axiom and some higher inductive types, as implemented in the cubical proof assistant. We show that the underlying cube category is the opposite of the Lawvere theory of De Morgan algebras. The topos of cubical sets itself classifies the theory of ‘free De Morgan algebras’...