. We extend the concept of constructive complete distributivity so as to make it applicable to ordered sets admitting merely bounded suprema. The KZ-doctrine for bounded suprema is of some independent interest and a few results about it are given. The 2-category of ordered sets admitting bounded suprema over which non-empty infima distribute is shown to be bi-equivalent to a 2-category defined in terms of idempotent relations. As a corollary we obtain a simple construction of the non-negative reals. 1. Introduction 1.1. The main theorem of [RW1] exhibited a bi-equivalence between the 2-category of (constructively) completely distributive lattices and sup-preserving arrows, and the idempotent splitting completion of the 2-category of relations --- relative to any base topos. Somewhat in passing in [RW1], it was pointed out that this bi-equivalence provides a simple construction of the closed unit interval ([0; 1]; ), namely as the ordered set of down-sets for the idempotent relat...

We pursue distributive laws between monads, particularly in the context of KZ-doctrines, and show that a very basic distributive law has (constructively) completely distributive lattices for its algebras. Moreover, the resulting monad is shown to be also the double dualization monad (with respect to the subobject classifier) on ordered sets.

The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.

ABSTRACT. By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a self-adjoint exponential Σ (−) on some category, is monadic. Using Beck’s theorem, this means that certain equalisers exist and carry the subspace topology. These subspaces are encoded by idempotents that play a role similar to that of nuclei in locale theory. Paré showed that any elementary topos has this duality, and we prove it intuitionistically for the category of locally compact locales. The paper is largely concerned with the construction of such a category out of one that merely has powers of some fixed object Σ. It builds on Sober Spaces and Continuations, where the related but weaker notion of abstract sobriety was considered. The construction is done first by formally adjoining certain equalisers that Σ (−) takes to coequalisers, then using Eilenberg–Moore algebras, and finally presented as a lambda calculus similar to the axiom of comprehension in set theory. The comprehension calculus has a normalisation theorem, by which every type can

The category of locally compact locales over any elementary topos is characterised by means of the axioms of abstract Stone duality (monadicity of the topology, considered as a selfadjoint exponential # , and Scott continuity, F# = ##.

The notion of Frobenius algebra originally arose in ring theory, but it is a fairly easy observation that this notion can be extended to arbitrary monoidal categories. But, is this really the correct level of generalisation?

Abstract. The 2-category of constructively completely distributive lattices is shown to be bidual to a 2-category of generalized orders that admits a monadic schizophrenic object biadjunction over the 2-category of ordered sets. 1.