## Studying repleteness in the category of cpos

### BibTeX

@MISC{Makkai_studyingrepleteness,

author = {Michael Makkai},

title = {Studying repleteness in the category of cpos},

year = {}

}

### OpenURL

### Abstract

We consider the notion of replete object in the category of directed complete partial orders and Scott-continuous functions, and show that, contrary to previous expectations, there are non-replete objects. The same happens in the case of ω-complete posets. Synthetic Domain Theory developed from an idea of Dana Scott: it is consistent with intuitionistic set theory that all functions between domains are continuous. He never wrote about this point of view explicitly, though he presented his ideas in many lectures also suggesting that the model offered by Kleene’s realizability was appropriate, and influenced various thesis works, e.g. [10,13,11,8,12], see also [14]. SDT can now be recognized as defining the “good properties ” required on a category C (usually, a topos with a dominance t: 1 ✲ Σ) in order to develop domain theory within a theory of sets. One of the problems addressed early in the theory was the identification of the sets to be considered as the Scott domains. As one would expect in a synthetic approach, the collection of these should be determined by the “good properties ” of the universe, in an intrinsic way. The best suggestion so far for such a collection comes from [6,15,5] and is that of repleteness. It is an orthogonality condition, see [2], and determines the replete objects of C as those which are completely recoverable from their properties detected by Σ. Say that A is replete (wrt. Σ) if it is orthogonal to all f: X ✲ Y in 1

### Citations

80 |
The effective topos
- Hyland
- 1982
(Show Context)
Citation Context ...and stable continuous maps—this category is the completion wrt. finite products Prodfin(L) of L. The dominance t: 1 ✲ Σ is as before. Effectively given domains are replete in the effective topos, see =-=[4]-=-, where Σ is the equivalence relation on numbers with two classes: the one which defines t: 1 ✲ Σ consists of the (codes of) Turing machines converging on input 0. 1 Replete cpos The results raise an ... |

67 |
Categories of continuous functors
- Freyd, Kelly
(Show Context)
Citation Context ...d by the “good properties” of the universe, in an intrinsic way. The best suggestion so far for such a collection comes from [6,15,5] and is that of repleteness. It is an orthogonality condition, see =-=[2]-=-, and determines the replete objects of C as those which are completely recoverable from their properties detected by Σ. Say that A is replete (wrt. Σ) if it is orthogonal to all f: X ✲ Y in 1 Researc... |

60 |
Realizability Toposes and Language Semantics
- Longley
- 1994
(Show Context)
Citation Context ...is point of view explicitly, though he presented his ideas in many lectures also suggesting that the model offered by Kleene’s realizability was appropriate, and influenced various thesis works, e.g. =-=[10,13,11,8,12]-=-, see also [14]. SDT can now be recognized as defining the “good properties” required on a category C (usually, a topos with a dominance t: 1 ✲ Σ) in order to develop domain theory within a theory of ... |

53 |
First steps in synthetic domain theory
- Hyland
- 1991
(Show Context)
Citation Context ...expect in a synthetic approach, the collection of these should be determined by the “good properties” of the universe, in an intrinsic way. The best suggestion so far for such a collection comes from =-=[6,15,5]-=- and is that of repleteness. It is an orthogonality condition, see [2], and determines the replete objects of C as those which are completely recoverable from their properties detected by Σ. Say that ... |

45 |
Continuity and Effectiveness in Topoi
- Rosolini
- 1986
(Show Context)
Citation Context ...is point of view explicitly, though he presented his ideas in many lectures also suggesting that the model offered by Kleene’s realizability was appropriate, and influenced various thesis works, e.g. =-=[10,13,11,8,12]-=-, see also [14]. SDT can now be recognized as defining the “good properties” required on a category C (usually, a topos with a dominance t: 1 ✲ Σ) in order to develop domain theory within a theory of ... |

34 |
The fixed point property in synthetic domain theory
- Taylor
- 1991
(Show Context)
Citation Context ...expect in a synthetic approach, the collection of these should be determined by the “good properties” of the universe, in an intrinsic way. The best suggestion so far for such a collection comes from =-=[6,15,5]-=- and is that of repleteness. It is an orthogonality condition, see [2], and determines the replete objects of C as those which are completely recoverable from their properties detected by Σ. Say that ... |

27 |
Realizability and Recursive Mathematics
- McCarty
- 1984
(Show Context)
Citation Context ...is point of view explicitly, though he presented his ideas in many lectures also suggesting that the model offered by Kleene’s realizability was appropriate, and influenced various thesis works, e.g. =-=[10,13,11,8,12]-=-, see also [14]. SDT can now be recognized as defining the “good properties” required on a category C (usually, a topos with a dominance t: 1 ✲ Σ) in order to develop domain theory within a theory of ... |

17 |
is not always sober
- Johnstone
- 1981
(Show Context)
Citation Context ...rly, conditions (ii) and (iii) may be rewritten, under (i), with “all n” instead of “infinitely many n”. Note that, in order to get (i), we shall use new points and conditions very similar to that in =-=[7]-=-, see (d) below. Hence, to obtain an isomorphism between the closed subsets, add new elements and clauses: (d) for each ℓ, i, elements (ℓ, i)n with • (ℓ, i)n < (ℓ, n) • ∨ ✍ n (ℓ, i)n = (ℓ, i) (e) for ... |

15 |
Program Verification in Synthetic Domain Theory
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- 1995
(Show Context)
Citation Context |

13 |
Domain Theory in Realizability Toposes
- Phoa
- 1990
(Show Context)
Citation Context |

11 | Two models of synthetic domain theory
- Fiore, Rosolini
- 1997
(Show Context)
Citation Context ...solini C such that Σf : ΣY ✲ X Σ is iso, i.e. for such f and all α there is a unique β such that the following diagram commutes X f ✲ Y • • • • ❅ ❅❅❅ α ❘ β . ❄ A. The following results are known, see =-=[6,3,1]-=-. The full subcategory of the replete objects is a reflective exponential ideal of C. It is the smallest such containing Σ. Scott domains and continuous functions are replete in the topos of presheave... |

9 | The S-replete construction
- Hyland, Moggi
- 1995
(Show Context)
Citation Context ...expect in a synthetic approach, the collection of these should be determined by the “good properties” of the universe, in an intrinsic way. The best suggestion so far for such a collection comes from =-=[6,15,5]-=- and is that of repleteness. It is an orthogonality condition, see [2], and determines the replete objects of C as those which are completely recoverable from their properties detected by Σ. Say that ... |

1 |
Epis need not be
- Lehmann, Pasztor
- 1982
(Show Context)
Citation Context ...ints newly added to C. So the first step is to add new points to C: for each pair ℓ@[i] < ℓ, add (ℓ, i) with the conditions that (c) ℓ@[i] < (ℓ, i) < ℓ. The poset thus obtained is exactly domain E in =-=[9]-=-; but, since it appears as the first step for our construction, we shall like to call it B0. The top part of B0 looks something like in Fig. 1. 3Makkai, Rosolini Proposition 1.3 Let A0 be the cpo on ... |

1 |
Letter to Wesley Phoa
- Scott
- 1989
(Show Context)
Citation Context ...y, though he presented his ideas in many lectures also suggesting that the model offered by Kleene’s realizability was appropriate, and influenced various thesis works, e.g. [10,13,11,8,12], see also =-=[14]-=-. SDT can now be recognized as defining the “good properties” required on a category C (usually, a topos with a dominance t: 1 ✲ Σ) in order to develop domain theory within a theory of sets. One of th... |