## The King's Buildings Edinburgh EH9 3JZ, Scotland

### BibTeX

@MISC{Fiore_theking's,

author = {Marcelo Fiore},

title = {The King's Buildings Edinburgh EH9 3JZ, Scotland},

year = {}

}

### OpenURL

### Abstract

We provide an internal characterization of the category!-Cpo of!-complete posets and!-continuous functions within the model H of SDT recently introduced by the authors. It follows that!-cpos lie between the two extreme synthetic notions of domain given by repleteness and well-completeness.

### Citations

194 |
Topos Theory
- Johnstone
- 1977
(Show Context)
Citation Context ...ad classifying total maps (i.e. the identity monad), the monad classifying partial maps de ned on a decidable subobject (i.e. the monad on 1+( )), and the monad classifying all partial maps� see e.g. =-=[15]-=-. An example that will illustrate at an elementary level what will follow is given by (an extension of) the bottom-adding monad ( )? |that classi es partial maps de ned on upper-closed subsets| on Pos... |

73 | Categories of continuous functors - Freyd - 1972 |

66 |
Realizability Toposes and Language Semantics
- Longley
- 1995
(Show Context)
Citation Context ... of the category of !-cpos in the model H, and use it to compare the notions of repleteness and wellcompleteness which aretwo extreme synthetic notions of domain respectively suggested in [12,26] and =-=[16,17]-=-. The paper is organized as follows. First, we recall some category-theoretic background material and some internal notions in a topos. Then, we recall some facts of Synthetic Domain Theory, and the p... |

63 |
Axiomatic Domain Theory in Categories of Partial Maps
- Fiore
- 1994
(Show Context)
Citation Context ...ernalizing possibly non-terminating computations is usually named lifting and it is axiomatized by requiring that the unit classi es (certain) partial maps that are closed under composition� see e.g. =-=[22,4]-=-. Precisely, the conditions are that the naturality diagrams for the unit are pullbacks (= the natural transformation is cartesian), and, in the situation 1 D _ _ ! B 1 # L1 2 # A�Fiore and Rosolini ... |

53 |
First steps in synthetic domain theory
- Hyland
- 1991
(Show Context)
Citation Context ...h respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a set-theoretic universe given by a topos [18]) appeared in the literature, e.g. in =-=[12,26,10,24,21,17]-=-. All these notions enjoy the properties needed in denotational semantics, e.g. closure under sums, products, and exponentials, the admission of xed-point operators, and the solution of recursive doma... |

49 | Sheaves in geometry and logic - LANE, MOERDIJK - 1992 |

38 |
The fixed point property in synthetic domain theory
- Taylor
- 1991
(Show Context)
Citation Context ...h respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a set-theoretic universe given by a topos [18]) appeared in the literature, e.g. in =-=[12,26,10,24,21,17]-=-. All these notions enjoy the properties needed in denotational semantics, e.g. closure under sums, products, and exponentials, the admission of xed-point operators, and the solution of recursive doma... |

21 | A uniform approach to domain theory in realizability models
- Longley, Simpson
- 1997
(Show Context)
Citation Context ...h respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a set-theoretic universe given by a topos [18]) appeared in the literature, e.g. in =-=[12,26,10,24,21,17]-=-. All these notions enjoy the properties needed in denotational semantics, e.g. closure under sums, products, and exponentials, the admission of xed-point operators, and the solution of recursive doma... |

17 | An extension of models of axiomatic domain theory to models of synthetic domain theory
- Fiore, Plotkin
- 1996
(Show Context)
Citation Context ...hat an object A is complete if it is internally orthogonal to c� itiswell-complete if LA is complete. As in a model of SDT the terminal object is well-complete, we have the following proposition, cf. =-=[17,7]-=-. Proposition 2.1 (ii) a well-complete object is complete, (i) Well-complete objects are closed under lifting, (iii) an object is well-complete if and only if it is orthogonal to all pullbacks of c id... |

12 | Two models of synthetic domain theory
- Fiore, Rosolini
- 1997
(Show Context)
Citation Context ...choice is preferable to others, mostly because of lack of distinguishing examples. In a recent paper the authors introduced two Grothendieck toposes, H and H 2, and studied them as models of SDT� see =-=[8]-=-. Each model respectively embeds the following standard categories of domains: the category of !-complete posets and !-continuous functions, and the category of !-complete posets with !-continuous pul... |

9 | The S-replete construction
- Hyland, Moggi
- 1995
(Show Context)
Citation Context ...f the replete, the well-complete, and the complete objects. Thanks to the universal characterization of the category of replete objects as the smallest full re ective exponential ideal contaning (see =-=[12,13]-=-), we know that Rep(H) =Rep(!-Cpo). Thus, for the topos H the internal notion of repleteness determines a good class of \standard" domains. All constructions on !-cpos used in denotational semantics a... |

8 |
A presentation of the initial lift algebra
- Jibladze
- 1997
(Show Context)
Citation Context ... natural number object N) as follows fp 2 N j8n 2 N: pn ) pn+1g > ! N � see [12]. 10Fiore and Rosolini The initial L-algebra is the smallest L-subalgebra of the (inverse of the) nal L-coalgebra, see =-=[14]-=- for an explicit description. In our situation, since L: H !Hpreserves non-empty connected colimits that are computed as in bL, the initial L-algebra LI ! I is obtained from the colimit of the diagram... |

7 | Enrichment and representation theorems for categories of domains and continuous functions
- Fiore
- 1996
(Show Context)
Citation Context ...1), and let 2 def = 1 + 1 and d def = [?� >]: 2 ! . For an object A, de ne the path relation p A > ! A A as the image of the map A d : A ! A 2 = A A. Consider the following notion, used implicitly in =-=[5,6]-=-: an object is path-transitive when it is internally orthogonal to the unique mediating map t in 1 ? ! > # A AAAAAAAU # ! _ H HHHHHHH ... t... L? R j 2: The name is justi ed by the fact that when an o... |

7 |
Notes on synthetic domain theory
- Rosolini
- 1995
(Show Context)
Citation Context ...ss). In all the various cases, it is made sure, one way or the other, that the chosen domains are closed under the action of the monad of computations. We refer the reader to the cited literature and =-=[25]-=- for more on the subject of SDT. Here we recall some of the basic de nitions and the facts we need in the paper. The underlying functor L of a lifting monad always has an initial algebra and a nal coa... |

5 |
Repleteness and the associated sheaf
- Bucalo, Rosolini
- 1998
(Show Context)
Citation Context ... Q QQQQQs X ! LA c I X U ! A This follows again from the completeness of L1. 2 F id # X # C 3 Repleteness The de nition of repleteness ts into a general scheme to produce re ective subcategories� see =-=[9,13,1]-=-. In a model of SDT, an object is replete [12,26] (with respect to ) if it is orthogonal to all those maps f : X ! Y such that f Y X : ! is an iso. Roughly speaking, the replete objects are those obje... |

5 |
Concrete quasitopoi
- Dubuc
(Show Context)
Citation Context ...are those A for which the diagonal mono A: A > ! A A is ::-closed. Basic properties about the above notions, and about the ::-topology in particular, are stated in the following two propositions� see =-=[3,18]-=- for the proofs. Proposition 1.3 (i) An object is j-separated if and only if the re ection into its associated j-sheaf is monic. (ii) A mono m: X > ! A is j-dense if and only if the associated j-sheaf... |

4 | An enrichment theorem for an axiomatisation of categories of domains and continuous functions
- Fiore
- 1997
(Show Context)
Citation Context ...1), and let 2 def = 1 + 1 and d def = [?� >]: 2 ! . For an object A, de ne the path relation p A > ! A A as the image of the map A d : A ! A 2 = A A. Consider the following notion, used implicitly in =-=[5,6]-=-: an object is path-transitive when it is internally orthogonal to the unique mediating map t in 1 ? ! > # A AAAAAAAU # ! _ H HHHHHHH ... t... L? R j 2: The name is justi ed by the fact that when an o... |

3 | editors, Category Theory 90, volume 1488 - Pedicchio, Rosolini - 1990 |

3 |
Continuity and E ectiveness in Topoi
- Rosolini
- 1986
(Show Context)
Citation Context ...ernalizing possibly non-terminating computations is usually named lifting and it is axiomatized by requiring that the unit classi es (certain) partial maps that are closed under composition� see e.g. =-=[22,4]-=-. Precisely, the conditions are that the naturality diagrams for the unit are pullbacks (= the natural transformation is cartesian), and, in the situation 1 D _ _ ! B 1 # L1 2 # A�Fiore and Rosolini ... |

3 |
An exper model for quest
- Rosolini
- 1992
(Show Context)
Citation Context |

1 |
Studying repleteness in the category of cpos. This volume
- Makkai, Rosolini
- 1997
(Show Context)
Citation Context ...H (or in !-Cpo for that matter) coincides with some existent notion of domain: we only know that the inclusion is strict because of a counterexample worked out by Michael Makkai and the second author =-=[19]-=-. It is clear that the other inclusion !-Cpo ! WC(H) is strict by Theorem 4.4 as it is easy to imagine examples of complete objects which are not path-transitive or::-separated. In other words, one ca... |

1 |
Program Veri cation in Synthetic Domain Theory
- Reus
- 1995
(Show Context)
Citation Context |

1 |
Categories and e ective computations
- Rosolini
- 1987
(Show Context)
Citation Context ...etween domainlike objects are intrinsically continuous. In the literature there are at least four related suggestions for selecting the objects in the topos which are to be thought of as domains� see =-=[10,12,13,16,17,23,24,26]-=-. They all concentrate on abstracting peculiar properties of : they range from asking as many as possible (repleteness) to as few as acceptable (well-completeness). In all the various cases, it is mad... |