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A new notation for arrows
 In International Conference on Functional Programming (ICFP ’01
, 2001
"... The categorical notion of monad, used by Moggi to structure denotational descriptions, has proved to be a powerful tool for structuring combinator libraries. Moreover, the monadic programming style provides a convenient syntax for many kinds of computation, so that each library defines a new sublang ..."
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Cited by 52 (1 self)
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The categorical notion of monad, used by Moggi to structure denotational descriptions, has proved to be a powerful tool for structuring combinator libraries. Moreover, the monadic programming style provides a convenient syntax for many kinds of computation, so that each library defines a new sublanguage. Recently, several workers have proposed a generalization of monads, called variously “arrows ” or Freydcategories. The extra generality promises to increase the power, expressiveness and efficiency of the embedded approach, but does not mesh as well with the native abstraction and application. Definitions are typically given in a pointfree style, which is useful for proving general properties, but can be awkward for programming specific instances. In this paper we define a simple extension to the functional language Haskell that makes these new notions of computation more convenient to use. Our language is similar to the monadic style, and has similar reasoning properties. Moreover, it is extensible, in the sense that new combining forms can be defined as expressions in the host language. 1.
Modelling environments in callbyvalue programming languages
, 2003
"... In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environmen ..."
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Cited by 14 (4 self)
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In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environments in callbyvalue programming languages. There have been two more general definitions along both of these lines: the first generalising from cartesian to symmetric premonoidal categories, the second generalising from indexed categories with specified structure to κcategories. In this paper, we investigate environments in callbyvalue languages by analysing a finegrain variant of Moggi’s computational λcalculus, giving two equivalent sound and complete classes of models: one given by closed Freyd categories, which are based on symmetric premonoidal categories, the other given by closed κcategories.
Monads Need Not Be Endofunctors
"... Abstract. We introduce a generalisation of monads, called relative monads, allowing for underlying functors between different categories. Examples include finitedimensional vector spaces, untyped and typed λcalculus syntax and indexed containers. We show that the Kleisli and EilenbergMoore constr ..."
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Cited by 11 (2 self)
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Abstract. We introduce a generalisation of monads, called relative monads, allowing for underlying functors between different categories. Examples include finitedimensional vector spaces, untyped and typed λcalculus syntax and indexed containers. We show that the Kleisli and EilenbergMoore constructions carry over to relative monads and are related to relative adjunctions. Under reasonable assumptions, relative monads are monoids in the functor category concerned and extend to monads, giving rise to a coreflection between monads and relative monads. Arrows are also an instance of relative monads. 1
Under consideration for publication in Math. Struct. in Comp. Science Classical Linear Logic of Implications
, 2003
"... We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s dualcontext system for the intuitionistic case. The calculus has the nonlinear and linear implications as the basic constructs, and this design choice allows a te ..."
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We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkin’s dualcontext system for the intuitionistic case. The calculus has the nonlinear and linear implications as the basic constructs, and this design choice allows a technically manageable axiomatization without commuting conversions. Despite this simplicity, the calculus is shown to be sound and complete for categorytheoretic models given by ∗autonomous categories with linear exponential comonads. 1.
Coherence for SkewMonoidal Categories
"... I motivate a variation (due to K. Szlachányi) of monoidal categories called skewmonoidal categories where the unital and associativity laws are not required to be isomorphisms, only natural transformations. Coherence has to be formulated differently than in the wellknown monoidal case. In my (to ..."
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I motivate a variation (due to K. Szlachányi) of monoidal categories called skewmonoidal categories where the unital and associativity laws are not required to be isomorphisms, only natural transformations. Coherence has to be formulated differently than in the wellknown monoidal case. In my (to my knowledge new) version, it becomes a statement of uniqueness of normalizing rewrites. I present a proof of this coherence theorem and also formalize it fully in the dependently typed programming language Agda.
Submitted to: MSFP 2014 Coherence for SkewMonoidal Categories
"... We motivate a variation (due to K. Szlachányi) of monoidal categories called skewmonoidal categories where the unital and associativity laws are not required to be isomorphisms, only natural transformations. Coherence has to be formulated differently than in the wellknown monoidal case. In my (t ..."
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We motivate a variation (due to K. Szlachányi) of monoidal categories called skewmonoidal categories where the unital and associativity laws are not required to be isomorphisms, only natural transformations. Coherence has to be formulated differently than in the wellknown monoidal case. In my (to my knowledge new) version it becomes a statement of uniqueness of normalizing rewrites. We present a proof of this coherence proof and also formalize it fully in the dependently typed programming language Agda.