## Non-trivial Power Types can't be Subtypes of Polymorphic Types (1989)

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Venue: | in 4th Annual Symposium on Logic in Computer Science |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Pitts89non-trivialpower,

author = {Andrew M. Pitts},

title = {Non-trivial Power Types can't be Subtypes of Polymorphic Types},

booktitle = {in 4th Annual Symposium on Logic in Computer Science},

year = {1989},

pages = {6--13},

publisher = {IEEE Computer Society Press}

}

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### Abstract

This paper establishes a new, limitative relation between the polymorphic lambda calculus and the kind of higher-order type theory which is embodied in the logic of toposes. It is shown that any embedding in a topos of the cartesian closed category of (closed) types of a model of the polymorphic lambda calculus must place the polymorphic types well away from the powertypes oe !\Omega of the topos, in the sense that oe !\Omega is a subtype of a polymorphic type only in the case that oe is empty (and hence oe !\Omega is terminal) . As corollaries, we obtain strengthenings of Reynolds' result on the non-existence of settheoretic models of polymorphism. Introduction The results reported in this paper have their origin in Reynolds' discovery that the standard set-theoretic model of the simply typed lambda calulus cannot be extended to model the polymorphic, or second-order, typed lambda calculus. In [9] Reynolds speculated that there might be a model of polymorphism in which the typ...

### Citations

445 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...rtesian closed categories extends to a similar correspondence between theories in HOLT and toposes (which are those ccc's which also possess a subobject classifier). We refer the reader to Part II of =-=[6]-=- for a detailed account of this correspondence and for other, equivalent formulations of the higherorder logic of toposes. One of these equivalent formulations, and probably the most convenient one, i... |

175 |
The category-theoretic solution of recursive domain equations
- Smyth, Plotkin
(Show Context)
Citation Context ...rphisms (I�i) ;! (J�j) are morphisms f : I ;! J in E satisfying that j TP (f) =f i. It is well known that if (I�i)is an initial object in this category, then i is necessarily an isomorphism (see [5], =-=[13]-=-, [11]). (Recall that an object 0 in a category is initial if for every object X there is a unique morphism 0 ;! X� a weakly initial object satis es the same condition except for the uniqueness requir... |

58 |
A Fixpoint Theorem for Complete Categories
- Lambek
- 1968
(Show Context)
Citation Context ...ma! (J; j) are morphisms f : I \Gamma! J in E satisfying that j ffi T P (f) = f ffi i. It is well known that if (I; i) is an initial object in this category, then i is necessarily an isomorphism (see =-=[5]-=-, [13], [11]). (Recall that an object 0 in a category is initial if for every object X there is a unique morphism 0 \Gamma! X; a weakly initial object satisfies the same condition except for the uniqu... |

49 |
Polymorphism is set-theoretic constructively
- Pitts
- 1988
(Show Context)
Citation Context ...he particular topos Set which are ruled out by the non-constructive nature of classical set theory, become feasible for a more general topos. This is precisely the case for models of polymorphism. In =-=[8]-=- it was shown how to fully embed any categorical-style model of second-order typed lambda calculus in a topos in such a way that the original model appears in the corresponding internal logic of the t... |

38 |
Categorical semantics for higher order polymorphic lambda calculus
- Seely
- 1986
(Show Context)
Citation Context ...t: simple cardinality considerations show thatanysuch U would have tocontain only sets with at most one element. The categorical-style models, P, of polymorphism considered in [8] (and before that in =-=[12]-=-) are in particular K-models in the sense of Reynolds and Plotkin where K = P(1� U) is the ccc of (denotations of) closed types and terms in the model P. The construction of [8] results in a certain t... |

25 | The discrete objects in the effective topos
- Hyland, Rosolini
- 1990
(Show Context)
Citation Context ... As well as the examples manufactured in [8], one `naturally occuring ' example is the much-studied modest sets model of polymorphism, for which the enveloping topos is Hyland's effective topos : see =-=[3]-=- and [2]. But a non-trivial example of this kind of structure is not possible in the topos Set: simple cardinality considerations show that any such U would have to contain only sets with at most one ... |

18 |
Scott,Typed Lambda Models and Cartesian Closed Categories
- Mitchell, J
- 1989
(Show Context)
Citation Context ...ion for -abstraction and a limited form of fi-conversion for-abstraction) and is tailored to obtaining the results of that paper and no more. (See also [1] for a semantics in a similar style; and see =-=[7]-=- for a detailed comparison between the categorical- and the environment-style models in the case of the simply typed lambda calculus.) For both kinds of model, part of the structure is a cartesian clo... |

10 |
R.Meyer: The semantics of second order polymorphic lambda calculus
- Bruce
- 1984
(Show Context)
Citation Context ...onally quite weak (it satisfiessfi and j conversion for -abstraction and a limited form of fi-conversion for-abstraction) and is tailored to obtaining the results of that paper and no more. (See also =-=[1]-=- for a semantics in a similar style; and see [7] for a detailed comparison between the categorical- and the environment-style models in the case of the simply typed lambda calculus.) For both kinds of... |

1 |
A categorical approach to realizability and Page 7 polymorphic types
- Carboni, Freyd, et al.
- 1987
(Show Context)
Citation Context ... as the examples manufactured in [8], one `naturally occuring ' example is the much-studied modest sets model of polymorphism, for which the enveloping topos is Hyland's effective topos : see [3] and =-=[2]-=-. But a non-trivial example of this kind of structure is not possible in the topos Set: simple cardinality considerations show that any such U would have to contain only sets with at most one element.... |

1 | et al (eds), Semantics of Data Types - Kahn - 1984 |

1 |
A categorical approach to realizability and Page 7 types
- Carboni, Freyd, et al.
- 1987
(Show Context)
Citation Context ...ll as the examples manufactured in [8], one `naturally occuring' example is the much-studied modest sets model of polymorphism, for which the enveloping topos is Hyland's e ective topos : see [3] and =-=[2]-=-. But a non-trivial example of this kind of structure is not possible in the topos Set: simple cardinality considerations show thatanysuch U would have tocontain only sets with at most one element. Th... |

1 |
The discrete objects in the e ective topos
- Hyland, Robinson, et al.
(Show Context)
Citation Context ...s. As well as the examples manufactured in [8], one `naturally occuring' example is the much-studied modest sets model of polymorphism, for which the enveloping topos is Hyland's e ective topos : see =-=[3]-=- and [2]. But a non-trivial example of this kind of structure is not possible in the topos Set: simple cardinality considerations show thatanysuch U would have tocontain only sets with at most one ele... |