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43
Generic trace theory
 International Workshop on Coalgebraic Methods in Computer Science (CMCS 2006), volume 164 of Elect. Notes in Theor. Comp. Sci
, 2006
"... Trace semantics has been defined for various nondeterministic systems with different input/output types, or with different types of “nondeterminism ” such as classical nondeterminism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms ..."
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Trace semantics has been defined for various nondeterministic systems with different input/output types, or with different types of “nondeterminism ” such as classical nondeterminism (with a set of possible choices) vs. probabilistic nondeterminism. In this paper we claim that these various forms of “trace semantics” are instances of a single categorical construction, namely coinduction in a Kleisli category. This claim is based on our main technical result that an initial algebra in
Recursive Types in Kleisli Categories
 Preprint 2004. MFPS Tutorial, April 2007 Classical Domain Theory 75/75
, 1992
"... We show that an enriched version of Freyd's principle of versality holds in the Kleisli category of a commutative strong monad with fixedpoint object. This gives a general categorical setting in which it is possible to model recursive types involving the usual datatype constructors. ..."
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We show that an enriched version of Freyd's principle of versality holds in the Kleisli category of a commutative strong monad with fixedpoint object. This gives a general categorical setting in which it is possible to model recursive types involving the usual datatype constructors.
Selection Functions, Bar Recursion and Backward Induction
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE, 20, PP 127168
, 2010
"... ..."
Combining algebraic effects with continuations
, 2007
"... We consider the natural combinations of algebraic computational effects such as sideeffects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor ext ..."
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We consider the natural combinations of algebraic computational effects such as sideeffects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor extend, with effort, to include commonly used combinations of the various algebraic effects with continuations. Continuations also give rise to a third sort of combination, that given by applying the continuations monad transformer to an algebraic effect. We investigate the extent to which sum and tensor extend from algebraic effects to arbitrary monads, and the extent to which Felleisen et al.’s C operator extends from continuations to its combination with algebraic effects. To do all this, we use Dubuc’s characterisation of strong monads in terms of enriched large Lawvere theories.
Enrichment and Representation Theorems for Categories of Domains and Continuous Functions
, 1996
"... This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that Research supported b ..."
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This paper studies the notions of approximation and passage to the limit in an axiomatic setting. Our axiomatisation is subject to the following criteria: the axioms should be natural (so that they are available in as many contexts as possible) and nonordertheoretic (so that Research supported by SERC grant RR30735 and EC project Programming Language Semantics and Program Logics grant SC1000 795 they explain the ordertheoretic structure). Our aim is 1. to provide a justification of Scott's original consideration of ordered structures, and 2. to deepen our understanding of the notion of passage to the limit
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
Free Products of Higher Operad Algebras
, 909
"... Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, ..."
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Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product of 2categories. In this paper we continue the developments of [3] and [2] by understanding the natural generalisations of Gray’s little brother, the funny tensor product of categories. In fact we exhibit for any higher categorical structure definable by an noperad in the sense of Batanin [1], an analogous tensor product which forms a symmetric monoidal closed structure
Categories and Types for Axiomatic Domain Theory
, 2003
"... Domain Theory provides a denotational semantics for programming languages and calculi containing fixed point combinators and other socalled paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of D ..."
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Domain Theory provides a denotational semantics for programming languages and calculi containing fixed point combinators and other socalled paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of Domain Theory, we extend Benton’s linear/nonlinear dualsequent calculus to include recursive linear types and define a class of models by adding Freyd’s notion of algebraic compactness to the monoidal adjunctions that model Benton’s calculus. We observe that algebraic compactness is better behaved in the context of categories with structural actions than in the usual context of enriched categories. We establish a theory of structural algebraic compactness that allows us to describe our models without reference to enrichment. We develop a 2categorical perspective on structural actions, including a presentation of monoidal categories that leads directly to Kelly’s reduced coherence conditions. We observe that Benton’s adjoint type constructors can be treated individually, semantically as well as syntactically, using free representations of distributors. We type various of fixed point combinators using recursive types and function types, which
A.,Some problems and results in synthetic functional analysis
 in Category Theoretic Methods in Geometry, Proceedings Aarhus 1983, Aarhus Various Publication Series No
, 1983
"... This somewhat tentative note aims at making a status about “functional analysis ” in certain ringed toposes E, R, in particular, duality theory for Rmodules in E. This is an area where knowledge still seems fragmentary, but where some of the known facts indicate the importance of such theory. The f ..."
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This somewhat tentative note aims at making a status about “functional analysis ” in certain ringed toposes E, R, in particular, duality theory for Rmodules in E. This is an area where knowledge still seems fragmentary, but where some of the known facts indicate the importance of such theory. The first §, however, does not deal with specific toposes, but is of very general nature, making contact with the general theory of monads on closed categories, [3][6]. Most of the results and problems in this note were presented at the workshop. Closely related viewpoints were contained in Lawvere’s workshopcontribution on Intensive and Extensive Quantities, which have also influenced the presentation given in the following pages. 1 Restricted double dualization monads Let R be a ring in a topos E and let RMod denote the Ecategory of Rmodules (left Rmodules, say). The Evalued homfunctor for it is denoted Hom R(−, −). If X ∈ E and V ∈ RMod, there is a natural structure of Rmodule on V X = ΠXV, and for any U ∈ RMod Hom R(U, V X) ∼ = (Hom R(U, V)) X, naturally in X, U, and V. This expresses that V X is a cotensor of V with X (cf. e.g. [2]). Equivalently, it expresses that, for fixed V, we have a Estrong left adjoint V (−) : E → (RMod) op to the functor Hom R (−, V). Thus we
COMMUTATIVE MONADS AS A THEORY OF DISTRIBUTIONS
"... Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects cons ..."
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Abstract. It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and conditional probability.