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43
Notions of Computation Determine Monads
 Proc. FOSSACS 2002, Lecture Notes in Computer Science 2303
, 2002
"... We give semantics for notions of computation, also called computational effects, by means of operations and equations. We show that these generate several of the monads of primary interest that have been used to model computational effects, with the striking omission of the continuations monad, demo ..."
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Cited by 54 (7 self)
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We give semantics for notions of computation, also called computational effects, by means of operations and equations. We show that these generate several of the monads of primary interest that have been used to model computational effects, with the striking omission of the continuations monad, demonstrating the latter to be of a different character, as is computationally true. We focus on semantics for global and local state, showing that taking operations and equations as primitive yields a mathematical relationship that reflects their computational relationship.
Restriction categories I: Categories of partial maps
 Theoretical Computer Science
, 2001
"... ..."
Combining effects: sum and tensor
"... We seek a unified account of modularity for computational effects. We begin by reformulating Moggi’s monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations ..."
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Cited by 29 (4 self)
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We seek a unified account of modularity for computational effects. We begin by reformulating Moggi’s monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggi’s exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggi’s sideeffects monad transformer. Finally we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.
Variations on Algebra: monadicity and generalisations of equational theories
 Formal Aspects of Computing
, 2001
"... this paper the author was partially supported by an SERC/EPSRC Advanced Research Fellowship, EPSRC Research grant GR/L54639, and EU Working Group APPSEM ..."
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Cited by 25 (0 self)
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this paper the author was partially supported by an SERC/EPSRC Advanced Research Fellowship, EPSRC Research grant GR/L54639, and EU Working Group APPSEM
Combining Computational Effects: Commutativity and Sum
, 2002
"... We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi's paradigm for modelling computational effects; we emphasise the importance here of the operations that ..."
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Cited by 19 (4 self)
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We begin to develop a unified account of modularity for computational effects. We use the notion of enriched Lawvere theory, together with its relationship with strong monads, to reformulate Moggi's paradigm for modelling computational effects; we emphasise the importance here of the operations that induce computational effects. Effects qua theories are then combined by appropriate bifunctors (on the category of theories). We give a theory of the commutative combination of effects, which in particular yields Moggi's sideeffects monad transformer (an application is the combination of sideeffects with nondeterminism). And we give a theory...
An Axiomatic Approach to Binary Logical Relations with Applications to Data Refinement
 Proc. TACS'97, Springer LNCS 1281
, 1997
"... We introduce an axiomatic approach to logical relations and data refinement. We consider a programming language and the monad on the category of small categories generated by it. We identify abstract data types for the language with sketches for the associated monad, and define an axiomatic notion o ..."
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Cited by 18 (1 self)
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We introduce an axiomatic approach to logical relations and data refinement. We consider a programming language and the monad on the category of small categories generated by it. We identify abstract data types for the language with sketches for the associated monad, and define an axiomatic notion of "relation" between models of such a sketch in a semantic category. We then prove three results: (i) such models lift to the whole language together with the sketch; (ii) any such relation satisfies a soundness condition, and (iii) such relations compose. We do this for both equality of data representations and for an ordered version. Finally, we compare our formulation of data refinement with that of Hoare. This work has been done with the support of the MITI Cooperative Architecture Project. This author also acknowledges the support of Kakenhi. y This author achnowledges the support of the MITI Cooperative Architecture Project. z This author acknowledges the support of EPSRC grant...
Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 18 (8 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
Lax Logical Relations
 In 27th Intl. Colloq. on Automata, Languages and Programming, volume 1853 of LNCS
, 2000
"... Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambdacalculus terms. We show that lax logical relations coincide with th ..."
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Cited by 15 (2 self)
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Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambdacalculus terms. We show that lax logical relations coincide with the correspondences of Schoett, the algebraic relations of Mitchell and the prelogical relations of Honsell and Sannella on Henkin models, but also generalise naturally to models in cartesian closed categories and to richer languages.
Enriched Lawvere Theories
"... We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Morever, the Vcategory of mod ..."
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Cited by 15 (0 self)
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We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Morever, the Vcategory of models of a Lawvere Vtheory is equivalent to the Vcategory of algebras for the corresponding Vmonad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where V is Cat, and explain how the correspondence extends to pseudo maps of algebras.