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27
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 45 (5 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.
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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Cited by 17 (4 self)
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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.
Pseudodistributive laws
, 2004
"... We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudodistributive law between pseudomonads, and ..."
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Cited by 13 (1 self)
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We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudodistributive law between pseudomonads, and we show how the definition and the main theorems about it may be used to model several such structures simultaneously. Specifically, we address the relationship between pseudodistributive laws and the lifting of one pseudomonad to the 2category of algebras and to the Kleisli bicategory of another. This, for instance, sheds light on the preservation of some structures but not others along the Yoneda embedding. Our leading examples are given by the use of open maps to model bisimulation and by the logic of bunched implications.
Resource modalities in tensor logic
"... The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more ..."
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The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more primitive than linear logic. This revised point of view leads us to introduce tensor logic, a primitive variant of linear logic where negation is not involutive. After formulating its categorical semantics, we interpret tensor logic in a model based on Conway games equipped with a notion of payoff, in order to reflect the various resource policies of the logic: linear, affine, relevant or exponential.
CLOSED CATEGORIES VS. CLOSED MULTICATEGORIES
"... Abstract. We prove that the 2category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2subcategory of the 2category of closed multicategories. ..."
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Abstract. We prove that the 2category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2subcategory of the 2category of closed multicategories.
A unified categorytheoretic formulation of typed binding signatures
 In MERLIN’05
, 2005
"... We generalise Fiore et al's account of variable binding for untyped cartesian contexts and Tanaka's account of variable binding for untyped linear contexts to give an account of variable binding for simply typed axiomatically dened contexts. In line with earlier work by us, we axiomatise ..."
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We generalise Fiore et al's account of variable binding for untyped cartesian contexts and Tanaka's account of variable binding for untyped linear contexts to give an account of variable binding for simply typed axiomatically dened contexts. In line with earlier work by us, we axiomatise the notion of context by means of a pseudomonad S on Cat: Fiore et al implicitly used the pseudomonad Tfp for small categories with nite products, and Tanaka implicitly used the pseudomonad Tsm for small symmetric monoidal categories. But here we also extend from untyped variable binding to typed variable binding. Given a set A of types, this involves generalising from Fiore et al's use of [; Set] to [(SA)op; SetA]. We dene a substitution monoidal structure on [(SA)op; SetA], give a denition of binding signature at this level of generality, and extend initial algebra semantics to this typed, axiomatic setting. This generalises and axiomatises previous work by Fiore et al and later authors in particular cases. In particular, it includes the Logic of Bunched Implications and variants, and it yields an improved axiomatic denition of binding signature even in the case of untyped binders.
Tensor products of finitely cocomplete and abelian categories
 Journal of Algebra
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Tensor product of symmetric monoidal categories
"... We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2cells monoidal natural transformations. Our tensor product together with a suitable ..."
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We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2cells monoidal natural transformations. Our tensor product together with a suitable unit is part of a structure on SMC that is a 2categorical version of the symmetric monoidal closed categories. This structure is surprisingly simple. In particular the arrows involved in the associativity and symmetry laws for the tensor and in the unit cancellation laws are 2natural and satisfy coherence axioms which are strictly commuting diagrams. We also show that the category quotient of SMC by the congruence generated by its 2cells admits a symmetric monoidal closed structure. 1 Summary of results Thomason’s famous result claims that symmetric monoidal categories model all connective spectra [Tho95]. The discovery of a symmetric monoidal structure on the category of structured spectra [EKMM97] suggests that a similar structure should exist on an adequate category with symmetric monoidal categories as objects. The first aim of this work is to give a reasonable candidate for a
Enrichments over symmetric Picard categories
, 2009
"... Categorical rings were introduced in [JiPi07], which we call 2rings. In these notes we present basic definitions and results regarding 2modules. This is work in progress. 1 ..."
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Categorical rings were introduced in [JiPi07], which we call 2rings. In these notes we present basic definitions and results regarding 2modules. This is work in progress. 1