## Freyd is Kleisli, for arrows (2006)

Venue: | In C. McBride, T. Uustalu, Proc. of Wksh. on Mathematically Structured Programming, MSFP 2006, Electron. Wkshs. in Computing. BCS |

Citations: | 4 - 2 self |

### BibTeX

@INPROCEEDINGS{Jacobs06freydis,

author = {Bart Jacobs and Ichiro Hasuo},

title = {Freyd is Kleisli, for arrows},

booktitle = {In C. McBride, T. Uustalu, Proc. of Wksh. on Mathematically Structured Programming, MSFP 2006, Electron. Wkshs. in Computing. BCS},

year = {2006}

}

### OpenURL

### Abstract

Arrows have been introduced in functional programming as generalisations of monads. They also generalise comonads. Fundamental structures associated with (co)monads are Kleisli categories and categories of (Eilenberg-Moore) algebras. Hence it makes sense to ask if there are analogous structures for Arrows. In this short note we shall take first steps in this direction, and identify for instance the Freyd

### Citations

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174 | Generalising monads to arrows - Hughes - 2000 |

130 |
The Formal Theory of Monads
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- 1972
(Show Context)
Citation Context ...estion: are there analogous constructions for Arrows? In this note we shall give first answers, which are formulated at a high level of abstraction, touching upon a 2-categorical description (like in =-=[11, 9]-=-). Concrete examples are missing at this stage— apart from examples that come from (co)monads. What we do is formulate notions of algebra and Kleisli category for an Arrow, and illustrate that they ar... |

102 | Premonoidal categories and notions of computation
- Power, Robinson
- 1997
(Show Context)
Citation Context ... parsers. See also [7, 5, 13, 1] for more applications. From a categorical point of view, Arrows have long been considered as an alternative formulation of the (older) concept of Freyd category, from =-=[10]-=-. Recently, an alternative categorical description of Arrows has emerged [3], as a monoid in a category of bifunctors C op × C → Sets. This description is closer to the functional one, and enables a p... |

52 | A new notation for arrows
- Paterson
- 2001
(Show Context)
Citation Context ...isli category, Algebra 1. INTRODUCTION Arrows have been introduced by Hughes [4] as generalisation of monads, in order to capture a common interface for non-trivial programs such as parsers. See also =-=[7, 5, 13, 1]-=- for more applications. From a categorical point of view, Arrows have long been considered as an alternative formulation of the (older) concept of Freyd category, from [10]. Recently, an alternative c... |

25 | Polytypic data conversion programs
- Jansson, Jeuring
- 2002
(Show Context)
Citation Context ...isli category, Algebra 1. INTRODUCTION Arrows have been introduced by Hughes [4] as generalisation of monads, in order to capture a common interface for non-trivial programs such as parsers. See also =-=[7, 5, 13, 1]-=- for more applications. From a categorical point of view, Arrows have long been considered as an alternative formulation of the (older) concept of Freyd category, from [10]. Recently, an alternative c... |

19 | The essence of dataflow programming - Uustalu, Vene - 2005 |

18 |
Handbook of Categorical Algebra, volume 50, 51 and 52 of Encyclopedia of Mathematics. Cambridge Univ
- Borceux
- 1994
(Show Context)
Citation Context ...n >>> can also be seen as a natural transformation A ⊗ A → A, for the MSFP ’06 2sFreyd is Kleisli, for Arrows standard tensor product ⊗ of bifunctors / profunctors / distributors C op × C → Sets, see =-=[2]-=-. In this way one can understand the triple (A, arr, >>>) as a monoid in a category of bifunctors. Such a monoid thus consists of a a functor A: C op × C → Sets and natural transformations A ⊗ A >>> a... |

14 | Arrows, like monads, are monoids
- Heunen, Jacobs
- 2006
(Show Context)
Citation Context ...int of view, Arrows have long been considered as an alternative formulation of the (older) concept of Freyd category, from [10]. Recently, an alternative categorical description of Arrows has emerged =-=[3]-=-, as a monoid in a category of bifunctors C op × C → Sets. This description is closer to the functional one, and enables a precise formulation of the one-to-one relationship between Arrows and Freyd c... |

10 | There and back again: arrows for invertible programming
- Alimarine, Smetsers, et al.
- 2005
(Show Context)
Citation Context ...isli category, Algebra 1. INTRODUCTION Arrows have been introduced by Hughes [4] as generalisation of monads, in order to capture a common interface for non-trivial programs such as parsers. See also =-=[7, 5, 13, 1]-=- for more applications. From a categorical point of view, Arrows have long been considered as an alternative formulation of the (older) concept of Freyd category, from [10]. Recently, an alternative c... |

7 |
Combining a monad and a comonad. Theor
- Power, Watanabe
- 2002
(Show Context)
Citation Context ...estion: are there analogous constructions for Arrows? In this note we shall give first answers, which are formulated at a high level of abstraction, touching upon a 2-categorical description (like in =-=[11, 9]-=-). Concrete examples are missing at this stage— apart from examples that come from (co)monads. What we do is formulate notions of algebra and Kleisli category for an Arrow, and illustrate that they ar... |

6 | Environments, continuation semantics and indexed categories, in
- Power, Thielecke
- 1997
(Show Context)
Citation Context ...by elements of the set A(X, Y ). Identities and composition are given by arr and >>> in the obvious manner. We then have a functor JA : C → CA given by arr. This forms an instance of a Freyd category =-=[8]-=-. For example, the pre-monoidal structure ⊠ of CA comes from the operation first; the functor JA preserves pre-monoidal structures on-the-nose. This mapping A ↦→ (C JA → CA) from Arrows to Freyd categ... |

5 | Signals and comonads
- Uustalu, Vene
- 2005
(Show Context)
Citation Context |

2 | The essence of dataflow programming (short version - Uustalu, Vene - 2005 |

1 |
Handbook of Categorical Algebra, vols. 50, 51 and 52 of Encyclopedia of Mathematics. Cambridge Univ
- Borceux
- 1994
(Show Context)
Citation Context ...or the Workshop on Mathematically Structured Functional Programming, MSFP 2006 2sFreyd is Kleisli, for Arrows standard tensor product ⊗ of bifunctors / profunctors / distributors C op × C → Sets, see =-=[2]-=-. In this way one can understand the triple (A, arr, >>>) as a monoid in a category of bifunctors. Such a monoid thus consists of a a functor A: C op × C → Sets and natural transformations A ⊗ A >>> a... |