Combining a monad and a comonad

Power, John; Watanabe, Hiroshi

doi:10.1016/s0304-3975(01)00024-x

Cited by 72 publications

(71 citation statements)

References 7 publications

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“…(7)) hold if and only if (2) holds. Finally, by Corollary 3.11, [19], (3) is equivalent to (4). Similarly, by Corollary 5.11, [19], (5) is equivalent to (6).…”

Section: R) Is a Monoidal Category If And Only If (F G ϕ) Satisfiesmentioning

confidence: 90%

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Braided Mixed Datums and Their Applications on Hom-Quantum Groups

Zhang¹,

Dong²

2017

Glasgow Math. J.

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Abstract. In this paper, we mainly provide a categorical view on the braided structures appearing in the Hom-quantum groups. Let C be a monoidal category on which F is a bimonad, G is a bicomonad, and ϕ is a distributive law, we discuss the necessary and sufficient conditions for C G F (ϕ), the category of mixed bimodules to be monoidal and braided. As applications, we discuss the Hom-type (co)quasitriangular structures, the Hom-Yetter-Drinfeld modules, and the Hom-Long dimodules.2010 Mathematics Subject Classification. 16T25, 15W30.

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“…(7)) hold if and only if (2) holds. Finally, by Corollary 3.11, [19], (3) is equivalent to (4). Similarly, by Corollary 5.11, [19], (5) is equivalent to (6).…”

Section: R) Is a Monoidal Category If And Only If (F G ϕ) Satisfiesmentioning

confidence: 90%

“…Recall from [19] and [20], if ‫ރ‬ denotes any 2-category, then the following data forms the 2-category of monads, which is denoted by Mnd(‫:)ރ‬…”

Section: R) Is a Monoidal Category If And Only If (F G ϕ) Satisfiesmentioning

confidence: 99%

Braided Mixed Datums and Their Applications on Hom-Quantum Groups

Zhang¹,

Dong²

2017

Glasgow Math. J.

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“…To capture these interactions semantically requires the interaction of a graded monad and a graded comonad. A standard categorical technique for combining a (non-graded) monad and (non-graded) comonad uses a distributive law between them [7,45,50]. Inspired by this, we lift notions of distributive law to the graded setting.…”

Section: Computational Effects and Monadsmentioning

confidence: 99%

“…They are graded generalisations of the classical distributive laws of a comonad D over a monad T [45] (and vice versa) and involve nontrivial interactions between two kinds of grading given by a matched pair. We first focus on one of eight variations.…”

Section: Distributive Lawsmentioning

confidence: 99%

Combining effects and coeffects via grading

et al. 2016

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Effects and coeffects are two general, complementary aspects of program behaviour. They roughly correspond to computations which change the execution context (effects) versus computations which make demands on the context (coeffects). Effectful features include partiality, non-determinism, input-output, state, and exceptions. Coeffectful features include resource demands, variable access, notions of linearity, and data input requirements.The effectful or coeffectful behaviour of a program can be captured and described via type-based analyses, with fine grained information provided by monoidal effect annotations and semiring coeffects. Various recent work has proposed models for such typed calculi in terms of graded (strong) monads for effects and graded (monoidal) comonads for coeffects.Effects and coeffects have been studied separately so far, but in practice many computations are both effectful and coeffectful, e.g., possibly throwing exceptions but with resource requirements. To remedy this, we introduce a new general calculus with a combined effect-coeffect system. This can describe both the changes and requirements that a program has on its context, as well as interactions between these effectful and coeffectful features of computation. The effect-coeffect system has a denotational model in terms of effect-graded monads and coeffect-graded comonads where interaction is expressed via the novel concept of graded distributive laws. This graded semantics unifies the syntactic type theory with the denotational model. We show that our calculus can be instantiated to describe in a natural way various different kinds of interaction between a program and its evaluation context.

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“…A very good news is that although comonads alone do not cover clocked dataflow computation, a solution is still close at hand. General and causal partialstream functions turn out to be describable in terms of distributive combinations of a comonad and a monad considered, e.g., in [8,33]. For reasons of space, we will only discuss causal partial-stream functions as more relevant.…”

Section: Comonadic Semanticsmentioning

confidence: 99%

The Essence of Dataflow Programming

Uustalu

Vene

2006

Central European Functional Programming School

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Abstract. We propose a novel, comonadic approach to dataflow (streambased) computation. This is based on the observation that both general and causal stream functions can be characterized as coKleisli arrows of comonads and on the intuition that comonads in general must be a good means to structure context-dependent computation. In particular, we develop a generic comonadic interpreter of languages for context-dependent computation and instantiate it for stream-based computation. We also discuss distributive laws of a comonad over a monad as a means to structure combinations of effectful and context-dependent computation. We apply the latter to analyse clocked dataflow (partial stream based) computation.

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Combining a monad and a comonad

Cited by 72 publications

References 7 publications

Braided Mixed Datums and Their Applications on Hom-Quantum Groups

Braided Mixed Datums and Their Applications on Hom-Quantum Groups

Combining effects and coeffects via grading

The Essence of Dataflow Programming

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