## The Discrete Objects in the Effective Topos (1990)

Venue: | Proc. London Math. Soc |

Citations: | 25 - 6 self |

### BibTeX

@ARTICLE{Hyland90thediscrete,

author = {J. M. E. Hyland and E. P. Robinson and G. Rosolini},

title = {The Discrete Objects in the Effective Topos},

journal = {Proc. London Math. Soc},

year = {1990},

volume = {3}

}

### Years of Citing Articles

### OpenURL

### Abstract

The original aim of this paper was to give a rather quick and undemanding proof that the effective topos contains two non-trivial small (i.e. internal) full subcategories which are closed under all small limits in the topos (and hence in particular are internally complete). The interest in such subcategories arises from

### Citations

183 |
Topos Theory
- Johnstone
- 1977
(Show Context)
Citation Context ...phic lambdacalculi. A note on the effective topos Throughout the paper we shall refer to a topos called the effective topos, but as the reader familiar with the papers by Hyland, Johnstone, and Pitts =-=[12]-=- and by Hyland [10] will immediately realise, our results apply equally well to any realisability topos—a topos constructed from a partial applicative algebra in the same way that 'the' effective topo... |

141 |
The system F of variable types, fifteen years later. Theoretical Computer Science 45
- Girard
- 1986
(Show Context)
Citation Context ... be taken to be objects of the topos. Thus, if we take as the simplest real examplesTHE DISCRETE OBJECTS IN THE EFFECTIVE TOPOS 33 Girard's 'Systeme F' (Reynold's second-order lambda-calculus, Girard =-=[7]-=-, Reynolds [16]), we have type expressions which may only depend on variables ranging over types, whereas in the extension to Fw we have type expressions which depend on variables ranging over higher-... |

80 |
The effective topos
- Hyland
- 1982
(Show Context)
Citation Context ... A note on the effective topos Throughout the paper we shall refer to a topos called the effective topos, but as the reader familiar with the papers by Hyland, Johnstone, and Pitts [12] and by Hyland =-=[10]-=- will immediately realise, our results apply equally well to any realisability topos—a topos constructed from a partial applicative algebra in the same way that 'the' effective topos is constructed fr... |

72 |
Constructions: A Higher Order Proof System for Mechanizing Mathematics
- Coquand, Huet
- 1985
(Show Context)
Citation Context ...e the fact that P is not just a complete internal category, but a complete internal full subcategory of %fl? (or at least Sf^v) to obtain a model for the theory of constructions (cf. Coquand and Huet =-=[4]-=-). This theory can be seen as an extension of the full higher-order polymorphic lambda calculus more or less as follows: 1. the category of types (Coquand's and Huet's propositions) is indexed over it... |

67 |
Categories of continuous functors
- Freyd, Kelly
(Show Context)
Citation Context ...y. Everything we have to say in this section holds also for orthogonality with respect to such a family, but this generality will not be needed in the sequel. The reader familiar with Freyd and Kelly =-=[6]-=- will immediately realise that several of the arguments carried out in this section are the internal versions of results there, although we shall not produce them in their greatest generality. Let 0d&... |

48 |
Polymorphism is set theoretic, constructively
- Pitts
- 1987
(Show Context)
Citation Context ...ly, it is a version of the 2rAC-hyperdoctrines used by Pitts to prove a completeness result for internal category models with respect to the second-order lambda calculus. We refer the reader to Pitts =-=[15]-=- for a more detailed account of how to use internal categories than the sketch here. We shall indicate how a sufficiently complete internal category C in % can be used to model Systeme F extended by n... |

38 |
Fibred categories and the foundations of naive category theory
- Bénabou
- 1985
(Show Context)
Citation Context ... of this theory, and although in the interests of being self-contained we include the basic definitions, readers are referred to Grothendieck [9], Giraud [8] and, especially, to BSnabou's JSL article =-=[1]-=-. Let k and 3> be categories, p: si-* 3> a functor. A morphism a: Ax-+A0 of si is said to be cartesian with respect to p if for every )8: A2—>A0 and y: pA2—*pAx in 2) such that pa°y=pf} there is a uni... |

37 |
Categorical semantics for higher order polymorphic lambda calculus
- Seely
- 1986
(Show Context)
Citation Context ...ral way—as types varying over the interpretation of the domains of the free variables, and so as objects of [C] in the fibre over some /. For those familiar with the work of Seely in this area (Seely =-=[18]-=-), what we are essentially doing is using a restriction of the externalisation process to give a hyperdoctrine over some subcategory of the topos, and hence a version of Seely's notion of PL-category.... |

34 |
A small complete category
- Hyland
- 1987
(Show Context)
Citation Context ...quotients of N, and so to be the stack of separated families orthogonal to a2. Instead of {P} we shall sometimes write Mod, as it is the stack corresponding to the category of modest sets (cf. Hyland =-=[11]-=-). 7.3. PROPOSITION. The externalisation [P] is weakly but not strongly equivalent to Proof. The counter-example given at the end of § 6 for Proposition 6.9 serves equally well in this case. 7.4. CORO... |

32 | Abstract families and the adjoint functor theorem - Paré, Schumacher - 1978 |

12 |
Categories ´ Fibrees ´ et Descente
- Grothendieck
- 1971
(Show Context)
Citation Context ...include here. 1. Fibrations There is as yet no good reference for the theory of fibrations as it appears from the point of view of categorical logic: Grothendieck's development of the theory in, say, =-=[9]-=- was inspired by vastly different ends than those likely to concern readers of this paper, and the canonical reference on fibrations for the modern category theorist, Jean B6nabou's long-awaited book,... |

7 |
Polymorphism is not Set-Theoretic. in: Semantics of Data Types
- Reynolds
(Show Context)
Citation Context ... The nature of these models is that types are interpreted as objects of the topos, and hence as 'sets' in an intuitionistic sense. They in fact provide 'set-theoretic' models in the sense of Reynolds =-=[16]-=-, except that the set-theory involved is non-classical. At the end of the paper we indicate briefly how one (only) of the two full subcategories gives rise to a model of the theory of constructions. T... |

5 |
completions and Morita equivalence for categories in a topos, Cahiers de Top. et Géo
- Bunge, Stack
(Show Context)
Citation Context ... is the full subcategory in Sf^S 1 ) on the single object the connected double-covering of the circle. This double-covering is locally isomorphic to S 1 !^ 1 , but of course is not globally so. Bunge =-=[2]-=- shows how to construct, given a locally small fibration & over the topos <£, a stack 2F with a weak equivalence F: 2F-* & over % such that for any G: &-* <3 into a stack there is H: &^> <S such that ... |

4 |
Stacks and equivalence of indexed categories, Cahiers de Top. et Géo
- Bunge, Paré
(Show Context)
Citation Context ...equivalence in mathematics proceed by first establishing a weak equivalence in one direction and then using choice to obtain a pseudo-inverse. As has been observed by, for example, Bunge and Pare" in =-=[3]-=- (where the authors however use the language of indexed category theory), we can detect many interesting properties of an internal functor F at the level of its externalisation. 3.4. PROPOSITION. Supp... |

4 |
On canonizing category theory, or, on functorializing model theory
- Freyd
- 1974
(Show Context)
Citation Context ...transitive, and that it is therefore the closure as an equivalence relation of the relation on categories generated by the weak equivalence functors. This notion has been extensively treated by Freyd =-=[5]-=- who notes that many proofs of equivalence in mathematics proceed by first establishing a weak equivalence in one direction and then using choice to obtain a pseudo-inverse. As has been observed by, f... |

1 |
Cohomologie non abflienne
- GIRAUD
- 1971
(Show Context)
Citation Context ...global problem, since 'existence' in the internal logic of a topos corresponds to local existence. There is therefore a body of material emanating from the school of Grothendieck (for example, Giraud =-=[8]-=-), which is relevant to our present problems. Much of this is concerned with the linked notions of descent and of stack—a short section on which is included here. With a section on the general theory ... |

1 |
Church's thesis and a unification of types', paper presented at the conference 'Church's thesis after fifty years
- SCOTT
- 1986
(Show Context)
Citation Context ...small category of sets^—the internal category P of separated subquotients of N, which is the internal category corresponding to the subcategory of ^"called by Scott the category of modest sets (Scott =-=[17]-=-). Let Po be the object of closed partial equivalence relations on N, that is, Po= {S e Q^I-I| SeQo}. In the fibre over Po we have a generic closed partials30 J. M. E. HYLAND, E. P. ROBINSON AND G. RO... |