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Lifting as a KZdoctrine
 Proceedings of the 6 th International Conference, CTCS'95, volume 953 of Lecture Notes in Computer Science
, 1995
"... this paper, is the analysis of notions of approximation aiming at explaining and justifying (ordertheoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and orderenrichment we considered contextual approximation which, in ..."
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this paper, is the analysis of notions of approximation aiming at explaining and justifying (ordertheoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and orderenrichment we considered contextual approximation which, in the framework we were working in, coincided with the specialisation preorder . But in the applications carried out in [FP94, Fio94a] we had to work with an axiomatised notion of approximation, instead of the aforementioned one, for the following two reasons: first, the specialisation preorder is not appropriate in categories of domains and stable functions (see [Fio94c]) and, second, we do not know of nonordertheoretic axioms making the specialisation preorder !complete. To overcome these drawbacks another notion of approximation was to be considered. And, it was the second problem that motivated the intensional notion of approximation provided by the path relation. In fact, it is shown in [Fio94b] that under suitable axioms the path relation can be equipped with a canonical passagetothelimit operator appropriate for fixedpoint computations; stronger axioms make this operator be given by lubs of !chains
OrderEnrichment for Categories of Partial Maps
, 1993
"... Introduction In (Plotkin 1985) a revitalised approach to domain theory was initiated. Roughly, the idea was to eliminate the bottom from the domains and to keep the functions partially defined. Thus replacing Cppo (the category of small cppos posets with a least element and closed under lubs of ..."
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Introduction In (Plotkin 1985) a revitalised approach to domain theory was initiated. Roughly, the idea was to eliminate the bottom from the domains and to keep the functions partially defined. Thus replacing Cppo (the category of small cppos posets with a least element and closed under lubs of !chains and continuous functions) with pCpo (the category of small cpos posets closed under lubs of !chains and partial continuous functions  see Subsection 3.1). One important point in the reformulation is the recognition of pCpo Research partially supported by Fundaci'on Antorchas and The British Council grant ARG 2281/14/6, and SERC grant RR30735. as a category of partial maps as, for example, such presentation fits better with standard formulations of recursion theory and it allows a categorical description of data types (via partial cartesian closed categories (Longo and Moggi 1984) with finite coproducts) in the presence of fixedpoint operators. Following the main moti
Equational lifting monads
 Proceedings CTCS '99, Electronic Notes in Computer Science
, 1999
"... We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of ..."
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We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the (partial) equational properties of partial map classifiers. The representation theorem also provides a tool for transferring nonequational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of the Kleisli categories of equational lifting monads as abstract Kleisli categories with extra structure. This axiomatization is of interest in its own right. 1
Metric Spaces in Synthetic Topology
, 2010
"... We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and me ..."
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We investigate the relationship between the synthetic approach to topology, in which every set is equipped with an intrinsic topology, and constructive theory of metric spaces. We relate the synthetic notion of compactness of Cantor space to Brouwer’s Fan Principle. We show that the intrinsic and metric topologies of complete separable metric spaces coincide if they do so for Baire space. In Russian Constructivism the match between synthetic and metric topology breaks down, as even a very simple complete totally bounded space fails to be compact, and its topology is strictly finer than the metric topology. In contrast, in Brouwer’s intuitionism synthetic and metric notions of topology and compactness agree. 1
Partial Hyperdoctrines: Categorical Models for Partial Function Logic and Hoare Logic
, 1993
"... In this paper we provide a categorical interpretation of the firstorder Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the wellknown notion of a (firstorder) hyperdoctrine to include partial maps. The most important ..."
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In this paper we provide a categorical interpretation of the firstorder Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the wellknown notion of a (firstorder) hyperdoctrine to include partial maps. The most important new aspect of the resulting partial (first order) hyperdoctrine is a different notion of morphism between the fibres. We also use this partial hyperdoctrine to give a model for Beeson's Partial Function Logic such that (a version of) his axiomatization is complete w.r.t. this model. This shows the usefulness of the notion independent of its intended use as a model for Hoare logic.
An exact interpretation of while
 Theory and Formal Methods 1993: Proceedings of the First Imperial College, Department of Computing, Workshop on Theory and Formal Methods
, 1993
"... The behaviour and interaction of finite limits (products, pullbacks and equalisers) and colimits (coproducts and coequalisers) in the category of sets is illustrated in a “hands on ” way by giving the interpretation of a simple imperative language in terms of partial functions between sets of states ..."
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The behaviour and interaction of finite limits (products, pullbacks and equalisers) and colimits (coproducts and coequalisers) in the category of sets is illustrated in a “hands on ” way by giving the interpretation of a simple imperative language in terms of partial functions between sets of states. We show that the interpretation is a least fixed point and satisfies the usual proof rule for loop invariants.
Sheaf Toposes for Realizability
, 2001
"... We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over di#erent partial combinatory algebras. This research is part o ..."
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We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over di#erent partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott. 1