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17
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 73 (6 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.
Algebraic Operations and Generic Effects
 Applied Categorical Structures
, 2003
"... Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal Vcategory C with cotensors and a strong Vmonad T on C, we investigate axioms under which an ObCindexed family of operations of the form α_x : (Tx)^ν → (Tx)^ω provides ..."
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Cited by 48 (7 self)
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Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal Vcategory C with cotensors and a strong Vmonad T on C, we investigate axioms under which an ObCindexed family of operations of the form &alpha;_x : (Tx)^&nu; &rarr; (Tx)^&omega; provides semantics for algebraic operations on the computational &lambda;calculus. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. In particular, we define the notion of generic effect and show that to give a generic effect is equivalent to giving an algebraic operation. We further show how the usual monadic semantics of the computational &lambda;calculus extends uniformly to incorporate generic effects. We outline examples and nonexamples and we show that our definition also enriches one for callbyname languages with e#ects.
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 23 (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
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 ..."
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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...
Countable Lawvere Theories and Computational Effects
, 2006
"... Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere ..."
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Cited by 17 (3 self)
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Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.
Functorial boxes in string diagrams
, 2006
"... String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in prooftheory (like JeanYves Girard’s proofnets) and in concurrency theory (like Ro ..."
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Cited by 13 (3 self)
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String diagrams were introduced by Roger Penrose as a handy notation to manipulate morphisms in a monoidal category. In principle, this graphical notation should encompass the various pictorial systems introduced in prooftheory (like JeanYves Girard’s proofnets) and in concurrency theory (like Robin Milner’s bigraphs). This is not the case however, at least because string diagrams do not accomodate boxes — a key ingredient in these pictorial systems. In this short tutorial, based on our accidental rediscovery of an idea by Robin Cockett and Robert Seely, we explain how string diagrams may be extended with a notion of functorial box to depict a functor separating an inside world (its source category) from an outside world (its target category). We expose two elementary applications of the notation: first, we characterize graphically when a faithful balanced monoidal functor F: C − → D transports a trace operator from the category D
A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES
"... Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “l ..."
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Cited by 12 (1 self)
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Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous
Semantics for Algebraic Operations
 Proc. MFPS 17, Electronic Notes in Thoeret. Comp. Sci
, 2001
"... Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V category C with cotensors and a strong V monad T on C, we investigate axioms under which an ObC indexed family of operations of the form #x : (Tx) v # (Tx) w provides semantics for algebraic ope ..."
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Cited by 10 (2 self)
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Given a complete and cocomplete symmetric monoidal closed category V and a symmetric monoidal V category C with cotensors and a strong V monad T on C, we investigate axioms under which an ObC indexed family of operations of the form #x : (Tx) v # (Tx) w provides semantics for algebraic operations, which may be used to extend the usual monadic semantics of the computational #calculus uniformly. We recall a definition for which we have elsewhere given adequacy results, and we show that an enrichment of it is equivalent to a range of other possible natural definitions of algebraic operation. We outline examples and nonexamples and we show that our definition also enriches one for callbyname languages with e#ects. 1
Notions of Lawvere theory
"... Categorical universal algebra can be developed either using Lawvere theories (singlesorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of ..."
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Cited by 10 (0 self)
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Categorical universal algebra can be developed either using Lawvere theories (singlesorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of
Generic Models for Computational Effects
"... A Freydcategory is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in callbyvalue programming languages, such as the computational λcalculus, with computational effects. We develop the theory of Freydcategories with that in min ..."
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Cited by 6 (1 self)
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A Freydcategory is a subtle generalisation of the notion of a category with finite products. It is suitable for modelling environments in callbyvalue programming languages, such as the computational λcalculus, with computational effects. We develop the theory of Freydcategories with that in mind. We first show that any countable Lawvere theory, hence any signature of operations with countable arity subject to equations, directly generates a Freydcategory. We then give canonical, universal embeddings of Freydcategories into closed Freydcategories, characterised by being free cocompletions. The combination of the two constructions sends a signature of operations and equations to the Kleisli category for the monad on the category Set generated by it, thus refining the analysis of computational effects given by monads. That in turn allows a more structural analysis of the λccalculus. Our leading examples of signatures arise from sideeffects, interactive input/output and exceptions. We extend our analysis to an enriched setting in order to account for recursion and for computational effects and signatures that inherently involve it, such as partiality, nondeterminism and probabilistic nondeterminism. Key words: Freydcategory, enriched Yoneda embedding, conical colimit completion, canonical model