Fitch-style modal deduction, in which modalities are eliminated by opening a subordinate proof, and introduced by shutting one, were investigated in the 1990s as a basis for lambda calculi. We show that such calculi have good computational properties for a variety of intuitionistic modal logics. Semantics are given in cartesian closed categories equipped with an adjunction of endofunctors, with...

A continuous predicate on a domain, or more generally a topological space, can be concretely described as an open or closed set, or less obviously, as the set of all predicates consistent with it. Generalizing this scenario to quantitative predicates, we obtain under certain well-understood hypotheses an isomorphism between continuous functions on points and supremum preserving functions on ope...

In this paper we initiate the study of discrete random variables over domains. Our work is inspired by work of Daniele Varacca, who devised indexed valuations as models of probabilistic computation within domain theory. Our approach relies on new results about commutative monoids defined on domains that also allow actions of the non-negative reals. Using our approach, we define two such familie...

We introduce a new cartesian closed category of two-level arenas and innocent strategies to model intersection types that are refinements of simple types. Intuitively a property (respectively computation) on the upper level refines that on the lower level. We prove Subject Expansion—any lower-level computation is closely and canonically tracked by the upper-level computation that lies over it—w...

We introduce various notions of partial topos, i.e. “topos without terminal object”. The strongest one, called local topos, is motivated by the key examples of finite trees and sheaves with compact support. Local toposes satisfy all the usual exactness properties of toposes but are neither cartesian closed nor have a subobject classifier. Examples for the weaker notions are local homeomorphisms...

We apply the theory of generalised concrete data structures or gCDSs to construct a cartesian closed category of concrete array structures with explicit data layout The technical novelty is the array gCDS preserved by exponentiation whose isomorphisms relate higher order objects to their local parts This work is part of our search of semantic foundations for data parallel functional programming

Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call M-types. We derive existence results for M-types in locally cartesian closed pretoposes with a natural numbers object, using their inter...

There can be little doubt that Rational Choice Theory (RCT) with its emphasis on the 'instrumentally rational' individual as the foundation of the political process has significantly enhanced the scope of political science. This paper details many of the areas of political science in which our understanding of events has been significantly enhanced by the application of RCT. But, in the end, RC...

We show that a version of Martin-Löf type theory with extensional identity, a unit type N1,Σ,Π, and a base type is a free category with families (supporting these type formers) both in a 1and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We then show th...

We investigate the notion of a comodel of a (countable) Lawvere theory, an evident dual to the notion of model. By taking the forgetful functor from the category of comodels to Set, every (countable) Lawvere theory generates a comonad on Set. But while Lawvere theories are equivalent to finitary monads on Set, and that result extends to higher cardinality, no such result holds for comonads, and...