## An Equational Notion of Lifting Monad (2003)

Venue: | TITLE WILL BE SET BY THE PUBLISHER |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Bucalo03anequational,

author = {Anna Bucalo and Carsten Führmann and Alex Simpson},

title = {An Equational Notion of Lifting Monad},

booktitle = {TITLE WILL BE SET BY THE PUBLISHER},

year = {2003}

}

### OpenURL

### Abstract

We introduce the notion of an equational lifting monad: a commutative strong monad satisfying one additional equation (valid for monads arising from partial map classifiers). We prove that any equational lifting monad has a representation by a partial map classifier such that the Kleisli category of the former fully embeds in the partial category of the latter. Thus equational lifting monads precisely capture the equational properties of partial maps as induced by partial map classifiers. The representation theorem also provides a tool for transferring non-equational properties of partial map classifiers to equational lifting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right. 1

### Citations

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Citation Context ...ting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right. 1 Introduction Ever since Moggi's work =-=[13, 14]-=-, the use of strong monads has provided a structural discipline underpinning the categorical approach to denotational semantics. The underlying idea is to make a denotational distinction between the o... |

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Citation Context ...ting monads. It is proved using a direct axiomatization of Kleisli categories of equational lifting monads. This axiomatization is of interest in its own right. 1 Introduction Ever since Moggi's work =-=[13, 14]-=-, the use of strong monads has provided a structural discipline underpinning the categorical approach to denotational semantics. The underlying idea is to make a denotational distinction between the o... |

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Citation Context ...en an equational lifting monad is dominical (Theorem 3), and discuss other miscellaneous properties of lifting monads. 2 Preliminaries In this section, we briefly review facts we require about monads =-=[11, 1]-=-, monoidal categories [11], strong monads [10, 14] and idempotent splittings. The reader may prefer to skip this section, and refer back to it as and when necessary. First, some general remarks about ... |

189 | Toposes, Triples, and Theories
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Citation Context ...en an equational lifting monad is dominical (Theorem 3), and discuss other miscellaneous properties of lifting monads. 2 Preliminaries In this section, we briefly review facts we require about monads =-=[11, 1]-=-, monoidal categories [11], strong monads [10, 14] and idempotent splittings. The reader may prefer to skip this section, and refer back to it as and when necessary. First, some general remarks about ... |

67 |
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Citation Context ..., one obtains: id = L # L# = L # L# 1 # L##, id# = L# 1 # L(!) # t # # = L# 1 # t # #, L!# The left-hand diagram above expresses that any equational lifting monad is relevant in the sense of Jacobs [=-=9]-=-. (Not every commutative relevant monad is an equational lifting monad. A simple counterexample is the (-) 2 monad on Set.) The right-hand diagram has a couple of interesting consequences. One easy co... |

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Citation Context ...ing monads, by giving a direct axiomatization of the categorical structure of their Kleisli categories. This work continues in a tradition, exemplified by Robinson and Rosolini's notion of p-category =-=[16]-=-, of providing direct, domain-free axiomatizations of categories of partial maps. In the case of p-categories, the axiomatized categories correspond to categories of partial maps with a suitable produ... |

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Citation Context ...rem 3), and discuss other miscellaneous properties of lifting monads. 2 Preliminaries In this section, we briefly review facts we require about monads [11, 1], monoidal categories [11], strong monads =-=[10, 14]-=- and idempotent splittings. The reader may prefer to skip this section, and refer back to it as and when necessary. First, some general remarks about our policy towards structure-preserving functors b... |

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Citation Context ...n EU TMR Marie Curie Research Training Grant (Fuhrmann); and EPSRC Research Grant GR/K06109 (Simpson). 1 The study of such "lifting" monads goes back to work of Mulry, Rosolini and Moggi in =-=the 1980s [15, 17, 12]-=-. In particular, in his PhD thesis [17], Rosolini considered a general categorical approach to partiality (based on the associated notions of dominion, see Section 3, and dominance) and proved represe... |

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Citation Context ... in [16, p. 102], we associate a domain map A f # A to any map A f # B, by defining: f = A #id , f# # A# B # 1 # A The importance of domain maps is apparent throughout [16]. In fact, Cockett and Lack =-=[3]-=-, have recently based their restriction categories, which provide a very general axiomatization of categories of partial maps, entirely on equational properties of domain maps. From such properties, w... |

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Citation Context ...terest. One of the applications we have of the work in this paper is to establish properties of recursion in axiomatic domain theory. A general axiomatic analysis of recursion has been carried out in =-=[18]-=-, establishing equational completeness assuming the existence of su#ciently many final coalgebras. In the presence of an equational lifting monad, Kleisli exponentials and a (parameterized) natural nu... |

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Citation Context ...ion results into categories of partial functions on presheaf toposes. In computer science, this categorical approach to partiality has proved its value through applications in axiomatic domain theory =-=[5, 6]-=- and synthetic domain theory [8]. In spite of the above, there are reasons to look for more general approaches to partiality. In particular, the notion of dominion requires every partial map to have i... |

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Citation Context ...ion results into categories of partial functions on presheaf toposes. In computer science, this categorical approach to partiality has proved its value through applications in axiomatic domain theory =-=[5, 6]-=- and synthetic domain theory [8]. In spite of the above, there are reasons to look for more general approaches to partiality. In particular, the notion of dominion requires every partial map to have i... |

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14 |
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Citation Context ...n EU TMR Marie Curie Research Training Grant (Fuhrmann); and EPSRC Research Grant GR/K06109 (Simpson). 1 The study of such "lifting" monads goes back to work of Mulry, Rosolini and Moggi in =-=the 1980s [15, 17, 12]-=-. In particular, in his PhD thesis [17], Rosolini considered a general categorical approach to partiality (based on the associated notions of dominion, see Section 3, and dominance) and proved represe... |

11 |
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Citation Context ... Finally, we mention related work by Robin Cockett and Stephen Lack. Building on their still unpublished restriction categories [3], they have recently extended their axiomatization to lifting monads =-=[4]-=-. Their work nicely complements ours. We assume finite products in the underlying category, and emphasise equational properties, representation theorems and transference results. They characterise lif... |

8 |
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(Show Context)
Citation Context ...dition, is the Kleisli category of an equational lifting monad on its associated total category. Our axiomatization of such categories is obtained by extending the notion of abstract Kleisli category =-=[7]-=-, which provides a direct axiomatization of categories that arise as Kleisli categories, with the necessary additional structure, to obtain the notion of abstract Kleisli p-category. In Section 7 we c... |

8 |
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(Show Context)
Citation Context ...ugh we don't have a precise connection, it is worth remarking that equation (3) is reminiscent of the Euclidean principle, recently introduced by Taylor as part of a characterisation 24 of dominances =-=[19]-=-. It is also illuminating to consider the significance of (3) within Moggi's computational lambda-calculus [13]. It appears that equation (3) corresponds to the intersubstitutivity of e with x in the ... |

2 |
An Abstract Stone Duality, I: Geometric and Higher Order Logic
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(Show Context)
Citation Context ...urious. Although we don't have a precise connection, it is worth remarking that it is reminiscent of the Euclidean principle, recently introduced by Taylor as part of a characterisation of dominances =-=[18]-=-. It is also illuminating to consider the significance of equation (3) within Moggi's computational lambda-calculus [12]. It appears that equation (3) corresponds to the intersubstitutivity of e with ... |