## Equational lifting monads (1999)

Venue: | Proceedings CTCS '99, Electronic Notes in Computer Science |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Bucalo99equationallifting,

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

title = {Equational lifting monads},

booktitle = {Proceedings CTCS '99, Electronic Notes in Computer Science},

year = {1999},

publisher = {Elsevier}

}

### 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 (partial) equational properties of 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 the Kleisli categories of equational lifting monads as abstract Kleisli categories with extra structure. This axiomatization is of interest in its own right. 1

### Citations

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(Show Context)
Citation Context ...sli categories of equational lifting monads as abstract Kleisli categories with extra structure. This axiomatization is of interest in its own right. 1 Introduction Ever since Moggi’s pioneering work =-=[11,12]-=-, 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... |

201 | Toposes, Triples and Theories
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(Show Context)
Citation Context ...es the equaliser requirement if A η ✲ LA η ✲ Lη ✲ L 2 A is an equaliser diagram. (The terminology is due to Moggi [11]. A monad satisfying the equaliser requirement is also said to be of descent type =-=[1]-=-.) The mono requirement is easily seen to be equivalent to faithfulness of L and also to the faithfulness of the right-adjoint functor J from C to the Kleisli category CL. The equaliser requirement cl... |

69 |
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- 1992
(Show Context)
Citation Context ...(πLA) ◦ L(〈η, id〉) = L(π1) ◦ L(µ×!) ◦ t ◦ ∆ = L(π1) ◦ t ◦ (µ × L!) ◦ ∆ = L(π1) ◦ t ◦ 〈µ, L!〉 The left-hand diagram above expresses that any equational lifting monad is relevant in the sense of Jacobs =-=[8]-=-. (Not every commutative relevant monad is 8 (6)san equational lifting monad. A simple counterexample is the (−) 2 monad on the category of sets.) The right-hand diagram has a couple of interesting co... |

63 | Axiomatic Domain Theory in Categories of Partial Maps - Fiore - 1994 |

59 |
Categories of partial maps
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(Show Context)
Citation Context ...ting 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 (see [16] for other references). In the case of p-categories, the axiomatized categories correpond to categories of par... |

53 | First steps in synthetic domain theory - Hyland - 1991 |

47 | Continuity and effectiveness in topoi - Rosolini - 1986 |

33 | The partial lambda calculus
- Moggi
- 1988
(Show Context)
Citation Context ...partial functions. Thus, in a categorical setting, one looks for monads embodying notions of partiality. The study of such “lifting” monads goes back to work of Mulry, Rosolini and Moggi in the 1980s =-=[13,17,10]-=-. In particular, in his PhD thesis [17], Rosolini considered a general categorical approach to partiality (based on the associated notions of dominion, see Section 2, and dominance) and proved represe... |

17 | Cuboidal sets in axiomatic domain theory
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(Show Context)
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 =-=[4,5]-=- and synthetic domain theory [7]. However, 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 ... |

15 |
Generalized Banach-Mazur functionals in the topos of recursive sets
- Mulry
(Show Context)
Citation Context ...partial functions. Thus, in a categorical setting, one looks for monads embodying notions of partiality. The study of such “lifting” monads goes back to work of Mulry, Rosolini and Moggi in the 1980s =-=[13,17,10]-=-. In particular, in his PhD thesis [17], Rosolini considered a general categorical approach to partiality (based on the associated notions of dominion, see Section 2, and dominance) and proved represe... |

11 |
functors and monoidal monads, Archiv der Math.23
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(Show Context)
Citation Context ...(2) (3)sbelow into a pullback. id × η A × B id ✲ A × B ∩ ❄ A × LB ∩ η t ❄ ✲ L(A × B) Then A × LB t ✲ L(A × B) is natural in A and B and provides a (unique) strength for the monad in the sense of Kock =-=[9]-=- (see also [11,12]). The costrength LA ×B t′ ✲ L(A ×B) can be defined analogously. A related symmetric map LA × LB ψ ✲ L(A × B) is obtained as the unique morphism making the square below into a pullba... |

9 |
Direct models of the computational lambda-calculus
- Führmann
(Show Context)
Citation Context ...ion, 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 an abstract Kleisli category =-=[6]-=-, which provides a direct axiomatization of categories that arise as Kleisli categories, with the necessary additional structure. We call the axiomatized categories abstract Kleisli p-categories. In S... |

8 | Axioms and (counter)examples in synthetic domain theory
- Oosten, Simpson
(Show Context)
Citation Context ...er, although the above pullbacks exist (indeed Ass is finitely complete), this monad is not dominical. (Interestingly, the extension of L0 to a functor L : ModAss → ModAss on modified assemblies, see =-=[14]-=-, does yield a dominical lifting monad, providing a full and failthful dominical representation of L0.) Finally, we prove a curious fact whose importance has been impressed upon us by Paul Taylor. Pro... |

2 |
An Abstract Stone Duality, I: Geometric and Higher Order Logic
- Taylor
- 1999
(Show Context)
Citation Context ...tegory of sets, e.g. powerset (for which it fails) and exceptions, (−)+E, (for which it holds). We remark that equation (5) is somewhat similar to the Euclidean pinciple recently introduced by Taylor =-=[18]-=- as part of a characterisation of dominances. In the previous section we established that every dominical lifting monad is an equational lifting monad. The goal of this paper is twofold: to show that ... |

1 |
Properties of Fixed-points in Axiomatic Domain Theory. Abstract presented at FICS
- Plotkin, Simpson
- 1998
(Show Context)
Citation Context ...est. 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 33sof recursion has been carried out in =-=[15]-=-, establishing equational completeness assuming the existence of sufficiently many final coalgebras. In the presence of an equational lifting monad all the necessary final coalgebras can be constructe... |