## Lifting as a KZ-doctrine (1995)

Venue: | Proceedings of the 6 th International Conference, CTCS'95, volume 953 of Lecture Notes in Computer Science |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Fiore95liftingas,

author = {Marcelo P. Fiore},

title = {Lifting as a KZ-doctrine},

booktitle = {Proceedings of the 6 th International Conference, CTCS'95, volume 953 of Lecture Notes in Computer Science},

year = {1995},

pages = {146--158},

publisher = {SpringerVerlag}

}

### OpenURL

### Abstract

this paper, is the analysis of notions of approximation aiming at explaining and justifying (order-theoretic) properties of categories of domains. For example, in [Fio94c, Fio94a], while studying the interaction between partiality and order-enrichment we considered contextual approximation which, in the framework we were working in, coincided with the specialisation preorder . But in the applications carried out in [FP94, Fio94a] we had to work with an axiomatised notion of approximation, instead of the aforementioned one, for the following two reasons: first, the specialisation preorder is not appropriate in categories of domains and stable functions (see [Fio94c]) and, second, we do not know of non-order-theoretic axioms making the specialisation preorder !-complete. To overcome these drawbacks another notion of approximation was to be considered. And, it was the second problem that motivated the intensional notion of approximation provided by the path relation. In fact, it is shown in [Fio94b] that under suitable axioms the path relation can be equipped with a canonical passage-to-the-limit operator appropriate for fixed-point computations; stronger axioms make this operator be given by lubs of !-chains

### Citations

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Axiomatic Domain Theory in Categories of Partial Maps
- Fiore
- 1994
(Show Context)
Citation Context ... a given class of colimits. ? Research supported by SERC grant RR30735. The paper is organised as follows. In Section 2 we introduce the basic assumptions on our category of discourse. In the vein of =-=[Fio94a]-=-, we equip it with a subcategory of admissible monos allowing us to consider a notion of partiality from which the lifting monad arises. In Section 3 we consider the category of Eilenberg-Moore algebr... |

54 |
Topology: A Geometric Account of General Topology
- Brown
- 1988
(Show Context)
Citation Context ...only if there exists ff : \Sigma m \Theta A ! B such that ff : f m g.sRemark. Ideas similar to the above, though in the context of undirected paths, constitute the basics of homotopy theory (see e.g. =-=[Bro88]-=-). Convention. When the subscript is omitted in \Sigma m , ?m , ?m and m , it is assumed to be 1. For testing approximation, paths of length one suffice: Proposition 4.3 f v g if and only if f g. Proo... |

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

45 | Continuity and Effectiveness in Topoi - Rosolini - 1986 |

42 |
Monads on symmetric monoidal closed categories
- Kock
- 1970
(Show Context)
Citation Context ...(LJ; j; ) : C ! C where L(A f \Gamma! B) = [j A ; f ], j is cartesian (i.e. the squares required to commute by naturality are pullbacks), and A = [j A ffi jLA ; A]. Moreover, L is a commutative monad =-=[Koc70]-=- with tensorial strength t 0 : L(A) \Theta B ! L(A \Theta B) given by [j A \Theta B; A \Theta B]. Assumption 2 We assume C has binary products and exponentials, and p(C; D) has a zero object.sConventi... |

34 | An Axiomatisation of Computationally Adequate Domain-Theoretic Models of FPC - Fiore, Plotkin - 1994 |

34 | The fixed point property in synthetic domain theory - Taylor - 1991 |

13 | Domain Theory in Realizability Toposes - Phoa - 1990 |

10 | Categories of partial morphisms and the partial lambda-calculus - MOGGI - 1986 |

3 | editors. Category Theory - Carboni, Pedicchio, et al. - 1990 |

3 | Order-enrichment for categories of partial maps
- Fiore
- 1995
(Show Context)
Citation Context ...sed notion of approximation, instead of the aforementioned one, for the following two reasons: first, the specialisation preorder is not appropriate in categories of domains and stable functions (see =-=[Fio94c]-=-) and, second, we do not know of non-order-theoretic axioms making the specialisation preorder !-complete. To overcome these drawbacks another notion of approximation was to be considered. And, it was... |

3 | Algebras for the partial map classifier monad
- Kock
- 1991
(Show Context)
Citation Context ...iom under which it becomes a preorder is proposed. Afterwards, we study the interaction between the path relation and the lifting monad (sometimes called the partial map classifier monad ---see, e.g. =-=[Koc91]-=-) arising from the representability of partial maps. In particular, we provide a (non-order-theoretic) axiomatic framework in which the lifting monad becomes a KZ-doctrine with respect to the path rel... |

2 | Cpo-categories of partial maps - Fiore - 1993 |

1 |
First steps on the representation of domains. Manuscript (available from http://www.dcs.ed.ac.uk/home/mf/path.dvi
- Fiore
- 1994
(Show Context)
Citation Context ...acks another notion of approximation was to be considered. And, it was the second problem that motivated the intensional notion of approximation provided by the path relation. In fact, it is shown in =-=[Fio94b]-=- that under suitable axioms the path relation can be equipped with a canonical passage-to-the-limit operator appropriate for fixed-point computations; stronger axioms make this operator be given by lu... |

1 |
Monads for which structures are adjoints to units. Manuscript (to appear
- Kock
- 1994
(Show Context)
Citation Context ...ith respect to the path relation. We do this motivated by the fact that on posets lifting is the construction that freely adjoins suprema for all subsets with at most one element, whilst KZ-doctrines =-=[Koc94]-=- are equational 2-dimensional categorical algebras underlying the process of freely completing categories under a given class of colimits. ? Research supported by SERC grant RR30735. The paper is orga... |

1 |
Categorical structures. Manuscript (to appear in the Handbook of Algebra volume 2
- Street
- 1992
(Show Context)
Citation Context ... in Proposition 4.4 (2)) is a sesqui-category (i.e. a category enriched over CAT, the category of locally small categories and functors, with the funny symmetric monoidal closed structure ---see e.g. =-=[Str92]-=-).s10 Lifting as a KZ-doctrine We recall the definition of KZ-doctrine (when the 2-categorical structure is quasi-posetal): Definition 10.1 ([Koc94]) A KZ-doctrine on a Preo-category A consists of a P... |