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12
A Categorical Axiomatics for Bisimulation
 In Proc. of CONCUR’98, LNCS 1466
, 1998
"... We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in ..."
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Cited by 18 (8 self)
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We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in the Kleisli category for the 2monad T .
An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory
 In Proceedings of CSL 96
, 1997
"... . We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of ..."
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Cited by 17 (6 self)
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. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !inductive fixedpoint object (Definition 1...
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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Cited by 7 (2 self)
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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.
A Semantic Model for Graphical User Interfaces
, 2011
"... We give a denotational model for graphical user interface (GUI) programming in terms of the cartesian closed category of ultrametric spaces. The metric structure allows us to capture natural restrictions on reactive systems, such as causality, while still allowing recursively defined values. We capt ..."
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Cited by 7 (1 self)
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We give a denotational model for graphical user interface (GUI) programming in terms of the cartesian closed category of ultrametric spaces. The metric structure allows us to capture natural restrictions on reactive systems, such as causality, while still allowing recursively defined values. We capture the arbitrariness of user input (e.g., a user gets to decide the stream of clicks she sends to a program) by making use of the fact that the closed subsets of a metric space themselves form a metric space under the Hausdorff metric, allowing us to interpret nondeterminism with a “powerspace ” monad on ultrametric spaces. The powerspace monad is commutative, and hence gives rise to a model of linear logic. We exploit this fact by constructing a mixed linear/nonlinear domainspecific language for GUI programming. The linear sublanguage naturally captures the usage constraints on the various linear objects in GUIs, such as the elements of a DOM or scene graph. We have implemented this DSL as an extension to OCaml, and give examples demonstrating that programs in this style can be short and readable.
Involutive categories and monoids, with a GNScorrespondence
 In Quantum Physics and Logic (QPL
, 2010
"... This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of EilenbergMoore algebras of involutive monads are involutive, with conjugation for modules and vecto ..."
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Cited by 4 (2 self)
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This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of EilenbergMoore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the socalled GelfandNaimarkSegal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive symmetric monoidal categories. 1
Convexity, duality, and effects
 IFIP Theoretical Computer Science 2010, number 82(1) in IFIP Adv. in Inf. and Comm. Techn
, 2010
"... This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that we call ‘convex functors’; they include what are usually called probability distribution functors. T ..."
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Cited by 3 (2 self)
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This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that we call ‘convex functors’; they include what are usually called probability distribution functors. These relationships take the form of three adjunctions. Two of these three are ‘dual ’ adjunctions for convex sets, one time with the Boolean truth values {0, 1} as dualising object, and one time with the probablity values [0, 1]. The third adjunction is between effect algebras and convex functors. 1
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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Cited by 2 (2 self)
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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
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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Cited by 1 (0 self)
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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.
Towards a Categorical Axiomatics of Bisimulation
, 1999
"... We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. The axiomatisation centres on 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled u ..."
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We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. The axiomatisation centres on 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in the Kleisli category for the 2monad T . We may define the notion of open map for any such 2monad; in examples of interest, the definition agrees exactly with the usual notion of functional bisimulation. Under a condition on T , namely that it be a dense KZmonad, which we define, it follows that functors in Kl(T ) preserve open maps, i.e., they respect functional bisimulation. We further investigate structures on Kl(T ) that exist for axiomatic reasons, primarily because T is a dense KZmonad, and we study how those structures help to model operations on processes. We outline how this analysis gives ideas for modelling higher order processes. We conclude by making compariso...