Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally cartesian closed categories. We also show how to interpret rst-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic rst-order logic. As a consequence, a modied double-negation translation into type theory (without bracket types) is complete for all of classical rst-order logic.

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A pattern of interaction that arises again and again in programming, is a “handshake”, in which two agents exchange data. The exchange is thought of as provision of a service. Each interaction is initiated by a specific agent —the client or Angel, and concluded by the other —the server or Demon. We present a category in which the objects —called interaction structures in the paper — serve as descriptions of services provided across such handshaken interfaces. The morphisms —called (general) simulations— model components that provide one such service, relying on another. The morphisms are relations between the underlying sets of the interaction structures. The proof that a relation is a simulation can serve (in principle) as an executable program, whose specification is that it provides the service described by its domain, given an implementation of the service described by its codomain.

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this paper is necessarily an overview. I hope in the future to find the energy to write up a complete version, which will include all the details, arguments, connections and references omitted here. Also, if my exposition is somewhat ideological and my language nontechnical, it is because my goal is to take the initial steps towards a new conception.

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The theme of this paper is the relation between formal topology and the theory of domains. On one hand, domain theory can be seen as a branch of formal topology. On the other hand, the influence of domain theory on formal topology is twofold. Historically, the presence of the subset Con in Scott’s information systems has been the starting point for the introduction of the positivity predicate Pos in formal topology; also, the notion of approximable mapping has influenced the definition of continuous relations between formal topologies. Conceptually, since domain theory can be seen as a particular case, any notion and result in domain theory becomes a challenge for formal topology: how much of domain theory can be generalized to formal topology? My impression is that some open problems in one of the two fields could already have a solution in the other, and that is why an intensification of contact should be rewarding. 1 1. Formal topology What is formal topology? A good approximation to the correct answer is: formal topology is topology as developed in (Martin-Löf’s) type theory [3]. This means that it is intuitionistic and predicative. Actually, it is fully formalizable in an implementation of type theory, via what we have called the toolbox for subsets (cf. [7]); as a result, notation is quite standard, except for the use of ɛ, which is different from ∈, for elements of a subset U (which is a propositional function, and hence not a set: when S is a set and U ⊆ S, a ɛ U means that a ∈ S and U(a) is true). The adjective “formal ” is due to the stress on the pointfree approach to topology, to which one is naturally lead adopting type theory. The original main definition (cf. [4]) was: Definition 1.1 (1984-1987) A structure A = (S, ·, 1, ✁, Pos) is a formal topology when: 1 I am very grateful to the organizers of the workshop Domains IV, in particular to Dieter Spreen, for inviting me. 1 S is a set and (S, ·, 1) is a semilattice (or commutative monoid), called the base; a ✁ U prop (a ∈ S, U ⊆ S) is a cover, that is it satisfies reflexivity transitivity

this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for first-order theories can expressed in the framework of locales appears, for instance, in Fourman and Grayson [6], where the analogy between points of a locale and models of a theory is emphasised; the identification of formal points with Henkin sets, gives a precise form to this analogy. We replace the use of locales by formal topology, which can be expressed in a predicative framework such as Martin-Lof's type theory. Proof-theoretic issues are also considered by Dragalin [4], who presents a topological completeness proof using only finitary inductive definitions. Palmgren and Moerdijk [10] is also concerned with constructions of models: using sheaf semantics, they obtain a stronger conservativity result than the one in [3]. We will first investigate the difference between the Dedekind-MacNeille cover and the inductive cover. It easy to see that \Delta DM is stronger than \Delta I , that is, OE \Delta I U implies OE \Delta DM U , but the converse does not hold in general. The notion of point is not primitive in formal topology and therefore it is natural to require that a formal topology has some notion of positivity defined on the basic neighbourhoods; that a neighbourhood is positive then corresponds to, in ordinary point based topology, that it is inhabited by some point. We will show several negative results on positivity, both for the inductive topology and the Dedekind-MacNeille topology. The points of an inductive topology correspond to Henkin sets, but the Dedekind-MacNeille topology has, in general, no points. Our reasoning is constructi...

Abstract. We consider an extensional version, called qmTT, of the intensional Minimal Type Theory mTT, introduced in a previous paper with G. Sambin, enriched with proof-irrelevance of propositions and effective quotient sets. Then, by using the construction of total setoid à la Bishop we build a model of qmTT over mTT. The design of an extensional type theory with quotients and its interpretation in mTT is a key technical step in order to build a two level system to serve as a minimal foundation for constructive mathematics as advocated in the mentioned paper about mTT.