## Classifying Toposes for First Order Theories (1997)

Venue: | Annals of Pure and Applied Logic |

Citations: | 7 - 3 self |

### BibTeX

@TECHREPORT{Butz97classifyingtoposes,

author = {Carsten Butz and Peter T. Johnstone},

title = {Classifying Toposes for First Order Theories},

institution = {Annals of Pure and Applied Logic},

year = {1997}

}

### OpenURL

### Abstract

By a classifying topos for a first-order theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) first-order theory has a classifying topos in this sense, but we characterize those which do by an appropriate `smallness condition', and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every first-order theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heyting-valued completeness theorem for infinitary first-order logic.

### Citations

205 |
Topos theory
- Johnstone
- 1977
(Show Context)
Citation Context ...for any topos F , there is an equivalence between the category Top (F ; E) of geometric morphisms from F to E , and the category of flat and continuous functors C ! F (see also [16], section VII.7 or =-=[8]-=-, Proposition 2 7.13). Our aim in this section is to characterize those functors C ! F which correspond to open geometric morphisms, in the sense of [9] or [12]. Although, for most of this paper, our ... |

144 | Introduction to Higher-order Categorical Logic, Cambridge U - Lambek, Scott - 1986 |

125 |
An extension of the Galois theory of Grothendieck
- Joyal, Tierney
- 1984
(Show Context)
Citation Context ...ral characterizations of those geometric morphisms whose inverse image functors preserve arbitrary (infinitary) first-order logic: they are exactly the open geometric morphisms in the sense of [9] or =-=[12]-=-. So we might hope that, for at least some first-order theories T of interest, we could find a topos B fo (T) and a natural equivalence Open (F ; B fo (T)) ' T-Mod(F) ; where the left-hand side denote... |

50 |
Sheaves and logic
- Fourman, Scott
- 1979
(Show Context)
Citation Context ...o have a proper class of subobjects of 1. In the cases= !, 3.3 was proved by P. Freyd, around 1975; his proof may be found in [5] under the name `Stone representation theorem for logoi' (and see also =-=[4]-=-, theorem 5.22, where the theorem is stated in terms of models in categories of Heyting-valued sets). However, Freyd's proof (which admittedly yields a more precise result) is much less direct than th... |

44 | First order categorical logic - Makkai, Reyes - 1977 |

32 |
logique des topos
- Boileau, Joyal
- 1981
(Show Context)
Citation Context ...h member of the first set is deducible from the second, and vice versa. (Here it should be emphasized that our notion of deducibility is relative to an intuitionistic deduction-system such as that in =-=[2]-=----suitably extended to handle the infinitary connectives, cf. [18] or [13]---since we wish to study models of our theories in categories (such as non-Boolean toposes) where the rules of classical log... |

30 |
Proper maps of toposes
- Moerdijk, Vermeulen
(Show Context)
Citation Context ...ow several characterizations of those geometric morphisms whose inverse image functors preserve arbitrary (infinitary) first-order logic: they are exactly the open geometric morphisms in the sense of =-=[9]-=- or [12]. So we might hope that, for at least some first-order theories T of interest, we could find a topos B fo (T) and a natural equivalence Open (F ; B fo (T)) ' T-Mod(F) ; where the left-hand sid... |

9 |
Minimal models of Heyting arithmetic
- MOERDIJK, PALMGREN
- 1997
(Show Context)
Citation Context ...le below), and it turns out that this fourth condition is the one of greatest importance for us. Morphisms satisfying this condition seem to have been first explicitly considered (in the cases= !) in =-=[18]-=-, where they were called `weak elementary embeddings'; but this name seems unsatisfactory since such morphisms do not have to be monic. Accordingly, we introduce the name -elementary morphism of \Sigm... |

9 |
Constructive sheaf semantics
- PALMGREN
- 1997
(Show Context)
Citation Context ...r, Freyd's proof (which admittedly yields a more precise result) is much less direct than the above. The observation that y(UT ) yields a conservative model of T was made independently by E. Palmgren =-=[20]-=- and the first author. We note also that y(UT ) is a minimal model of T in the sense of Moerdijk and Palmgren [18]. Since any -Heyting functor is a -geometric functor, the equivalence of 2.4 yields a ... |

8 | On the non-existence of free complete Boolean algebras, Fund.Math
- Hales
- 1964
(Show Context)
Citation Context ...nitary first-order formulae, but of arbitrary (L1! ) infinitary ones as well. And there are simply too many of the latter, even if we restrict ourselves to propositional logic. It is well known ([6], =-=[7]-=-, [21]) that the free complete Boolean algebra on a countable infinity of generators is a proper class; perhaps less familiar is the fact [11] that the corresponding problem for complete Heyting algeb... |

6 |
Forcing and classifying topoi
- Scedrov
- 1984
(Show Context)
Citation Context ...at the generic model of a geometric theory may satisfy first-order sentences not derivable from that theory, is typical of all such theories having non-Boolean classifying toposes.) We recall that in =-=[1]-=- Blass and Scedrov characterized those coherent (that is, !-geometric) theories whose classifying toposes are Boolean: their characterization involved the conjunction of two conditions, of which the f... |

6 |
Change of base for toposes with generators
- Diaconescu
- 1975
(Show Context)
Citation Context ... was part of a project funded by the Netherlands Organization for Scientific Research NWO. 1 Open maps into sheaf toposes Let E be the topos of sheaves on a (small) site (C; J ). Diaconescu's theorem =-=[3]-=- asserts that, for any topos F , there is an equivalence between the category Top (F ; E) of geometric morphisms from F to E, and the category of flat and continuous functors C ! F (see also [15], sec... |

6 |
Universal projective geometry via topos theory
- Kock
(Show Context)
Citation Context ...n mathematical practice are (at least Morita-equivalent to) geometric ones, we do occasionally need to consider models of theories which are not geometric. (For example, as was first observed by Kock =-=[14]-=-, it can often be profitable to consider the non-geometric first-order sentences satisfied by the generic model of a geometric theory.) For such theories, we cannot hope to have a classifying topos in... |

3 | Infinite Boolean polynomials - Gaifman - 1964 |

2 |
Jongh: A class of intuitionistic connectives
- de
- 1980
(Show Context)
Citation Context ...t ourselves to propositional logic. It is well known ([6], [7], [21]) that the free complete Boolean algebra on a countable infinity of generators is a proper class; perhaps less familiar is the fact =-=[11]-=- that the corresponding problem for complete Heyting algebras occurs already with two generators---that is, there is a proper class of L1! formulae in two propositional variables, no two of which are ... |

2 |
Infinitary propositional intuitionistic logic, Notre Dame
- Kalicki
- 1980
(Show Context)
Citation Context ...ere it should be emphasized that our notion of deducibility is relative to an intuitionistic deduction-system such as that in [2]---suitably extended to handle the infinitary connectives, cf. [19] or =-=[13]-=----since we wish to study models of our theories in categories (such as non-Boolean toposes) where the rules of classical logic are not valid.) One might therefore be tempted to identify each theory w... |

2 |
Infinitary intuitionistic logic from a classical point of view
- Nadel
- 1978
(Show Context)
Citation Context ...ersa. (Here it should be emphasized that our notion of deducibility is relative to an intuitionistic deduction-system such as that in [2]---suitably extended to handle the infinitary connectives, cf. =-=[19]-=- or [13]---since we wish to study models of our theories in categories (such as non-Boolean toposes) where the rules of classical logic are not valid.) One might therefore be tempted to identify each ... |

1 |
New proof of a theorem of Gaifmann and
- Solovay
- 1966
(Show Context)
Citation Context ...y first-order formulae, but of arbitrary (L1! ) infinitary ones as well. And there are simply too many of the latter, even if we restrict ourselves to propositional logic. It is well known ([6], [7], =-=[21]-=-) that the free complete Boolean algebra on a countable infinity of generators is a proper class; perhaps less familiar is the fact [11] that the corresponding problem for complete Heyting algebras oc... |