## Presheaf models of constructive set theories (2004)

Citations: | 16 - 4 self |

### BibTeX

@MISC{Gambino04presheafmodels,

author = {Nicola Gambino},

title = {Presheaf models of constructive set theories},

year = {2004}

}

### OpenURL

### Abstract

Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic Set Theory and sketch an application to an independence result. 1. Variable sets in foundations and practice Presheaves are of central importance both for the foundations and the practice of mathematics. The notion of a presheaf formalizes well the idea of a variable set, that is relevant in all the areas of mathematics concerned with the study of indexed families of objects [19]. One may then readily see how presheaves are of interest also in foundations: both Cohen’s forcing models for classical set theories and Kripke models for intuitionistic logic involve the idea of sets indexed by stages. Constructive aspects start to emerge when one considers the internal logic of categories of presheaves. This logic, which does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects

### Citations

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(Show Context)
Citation Context ...ed predicative formal systems. By a generalised predicative formal system we mean here a system that is proof-theoretically reducible to Martin-Löf dependent type theories with W -types and universes =-=[20, 12]-=-. Generalised predicative systems typically contain axioms allowing generalized forms of inductive definitions [1] instead of proof-theoretically strong axioms such as Power Set. Our development will ... |

177 |
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Citation Context ...ally reducible to Martin-Löf dependent type theories with W -types and universes [20, 12]. Generalised predicative systems typically contain axioms allowing generalized forms of inductive definitions =-=[1]-=- instead of proof-theoretically strong axioms such as Power Set. Our development will focus on categories of classes rather than categories of sets as the starting point to define presheaves, thus ass... |

119 |
The type theoretic interpretation of constructive set theory
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Citation Context ...h to indicate how a category E with the properties of Definition 2.3 provides a categorical model for CST. The idea follows essentially from the setsas-trees interpretation of constructive set theory =-=[2, 3, 4]-=-. First, consider the small map π : W → V of axiom (S2) and its associated wellfounded tree Wπ. In the category CST, a map satisfying axiom (S2) can be defined by letting W =def {(x, y) | y ∈ x}, V = ... |

113 |
An extension of the Galois theory of Grothendieck
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(Show Context)
Citation Context ...oning [25]. Furthermore, it is sufficiently expressive to allow the definitions of complex mathematical constructions. This aspect has led to important developments in the study of elementary toposes =-=[16]-=-. The main purpose of this paper is to show how presheaves can be used to obtain models for constructive set theories [23, 5] analogous to the ones defined by Dana Scott for intuitionistic set theorie... |

112 | Categorical logic and type theory - Jacobs - 1999 |

88 | The HOL logic
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(Show Context)
Citation Context ... does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects and arrows, and can be used as an alternative to diagrammatic reasoning =-=[25]-=-. Furthermore, it is sufficiently expressive to allow the definitions of complex mathematical constructions. This aspect has led to important developments in the study of elementary toposes [16]. The ... |

49 | Constructive set theory - Myhill - 1975 |

46 | Notes on constructive set theory
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(Show Context)
Citation Context ...s aspect has led to important developments in the study of elementary toposes [16]. The main purpose of this paper is to show how presheaves can be used to obtain models for constructive set theories =-=[23, 5]-=- analogous to the ones defined by Dana Scott for intuitionistic set theories [26]. In order to do so, we will have to overcome the challenges intrinsic to working with generalised predicative formal s... |

41 |
A completeness theorem for open maps
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(Show Context)
Citation Context ... Our axioms are imposed on top of a standard group of axioms, those for a class of open maps, that are not recall here. These are axioms (A1) – (A7) of [15, Section 1.1], and were first formulated in =-=[14]-=-. The axioms presented below are tailored to construct models of CST. Axioms (S1), (S2) were introduced in [15, Definition 1.1], while (S3) was considered in [14, 6]. By a small object we mean an obje... |

38 |
Algebraic Set Theory
- Joyal, Moerdijk
- 1995
(Show Context)
Citation Context ...such as Power Set. Our development will focus on categories of classes rather than categories of sets as the starting point to define presheaves, thus assuming the perspective of Algebraic Set Theory =-=[15, 27, 7, 22, 6]-=-. The main reason for this choice is that the properties of categories of sets do not always reflect directly the set-theoretical axioms adopted to define them. There are indeed axioms, such as Replac... |

37 | Wellfounded trees in categories
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(Show Context)
Citation Context ...’s presheaf models fit into the paradigm of Algebraic Set Theory. The study of presheaf and sheaf categories in the generalised predicative setting was initiated by Ieke Moerdijk and Erik Palmgren in =-=[21, 22]-=-. They introduced the notion of stratified pseudotopos as a candidate for the notion of a predicative topos. A predicative topos should be a counterpart at the generalised predicative level of the not... |

35 | Type theories, toposes and constructive set theory: predicative aspects of AST
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- 2002
(Show Context)
Citation Context ...such as Power Set. Our development will focus on categories of classes rather than categories of sets as the starting point to define presheaves, thus assuming the perspective of Algebraic Set Theory =-=[15, 27, 7, 22, 6]-=-. The main reason for this choice is that the properties of categories of sets do not always reflect directly the set-theoretical axioms adopted to define them. There are indeed axioms, such as Replac... |

26 | Wellfounded trees and dependent polynomial functors
- Gambino, Hyland
- 2004
(Show Context)
Citation Context ...Note that a natural numbers object can be characterised as an initial algebra for the polynomial functor associated to either of the two canonical arrows 1 → 1 + 1. For more on wellfounded trees, see =-=[21, 11]-=-. The next definition isolates axioms for small maps corresponding to the properties of small maps for CST, as we will discuss after the definition. Our axioms are imposed on top of a standard group o... |

25 | The strength of some Martin-Löf type theories
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- 1994
(Show Context)
Citation Context ...ed predicative formal systems. By a generalised predicative formal system we mean here a system that is proof-theoretically reducible to Martin-Löf dependent type theories with W -types and universes =-=[20, 12]-=-. Generalised predicative systems typically contain axioms allowing generalized forms of inductive definitions [1] instead of proof-theoretically strong axioms such as Power Set. Our development will ... |

25 | Elementary axioms for categories of classes - Simpson - 1999 |

22 |
Sheaf models for set theory
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- 1980
(Show Context)
Citation Context ...ally an intuitionistic counterpart of classical Zermelo-Frankel set theory, is obtained from CST by adding Full Separation and Power Set [9]. Presheaf and sheaf models for IZF have been considered in =-=[26, 8]-=-. 2.2. Categories of classes. We now associate to the constructive set theory CST a category, called CST, and study how the axioms of CST determine the properties of CST. The category CST is defined a... |

21 |
The consistency of classical set theory relative to a set theory with intuitionistic logic
- Friedman
- 1973
(Show Context)
Citation Context ...CZF [17]. The intuitionistic set theory IZF which is is essentially an intuitionistic counterpart of classical Zermelo-Frankel set theory, is obtained from CST by adding Full Separation and Power Set =-=[9]-=-. Presheaf and sheaf models for IZF have been considered in [26, 8]. 2.2. Categories of classes. We now associate to the constructive set theory CST a category, called CST, and study how the axioms of... |

17 | Bernays-Gödel type theory
- Butz
(Show Context)
Citation Context ...such as Power Set. Our development will focus on categories of classes rather than categories of sets as the starting point to define presheaves, thus assuming the perspective of Algebraic Set Theory =-=[15, 27, 7, 22, 6]-=-. The main reason for this choice is that the properties of categories of sets do not always reflect directly the set-theoretical axioms adopted to define them. There are indeed axioms, such as Replac... |

9 |
Relating topos theory and set theory via categories of classes
- Awodey, Butz, et al.
- 2003
(Show Context)
Citation Context |

9 |
Sheaf interpretations for generalised predicative intuitionstic systems Doctoral Thesis
- Gambino
- 2002
(Show Context)
Citation Context ... φ, observing that the clauses defining the semantics of a restricted formula are themselves restricted. � Theorem 4.3. (Vπ, =) is a model of CST. Proof. The claim follows mainly from Theorem 6.17 in =-=[10]-=-, apart from the validity of Exponentiation, which is a consequence of Theorem 7.1 and Theorem 9.6 of [22]. � 5. Conclusions Since partially ordered sets are special small categories, presheaf models ... |

9 |
Coalgebras in a category of classes
- Warren
- 2007
(Show Context)
Citation Context ...f models, of developing their study, and of finding applications. All of these aspects will be considered here. Related work on categories of classes in constructive set theories is also presented in =-=[28]-=-. The approach to the construction of categorical models for constructive set theories taken there is slighly different from the one assumed here, even if both follow the perspective of Algebraic Set ... |

6 |
Independence results around constructive ZF
- Lubarsky
(Show Context)
Citation Context ...by replacing Subset Collection with Exponentiation [5]. Robert Lubarsky has recently proved that Subset Collection is independent of Exponentiation, thus showing that CST is a proper subsystem of CZF =-=[17]-=-. The intuitionistic set theory IZF which is is essentially an intuitionistic counterpart of classical Zermelo-Frankel set theory, is obtained from CST by adding Full Separation and Power Set [9]. Pre... |

6 |
Category-theoretic models for intuitionistic set theory. Manuscript slides of a talk given at Carnegie Mellon University (additions made in
- Scott
- 1998
(Show Context)
Citation Context ...The main purpose of this paper is to show how presheaves can be used to obtain models for constructive set theories [23, 5] analogous to the ones defined by Dana Scott for intuitionistic set theories =-=[26]-=-. In order to do so, we will have to overcome the challenges intrinsic to working with generalised predicative formal systems. By a generalised predicative formal system we mean here a system that is ... |

2 | Constructive completions of ordered sets, groups and fields
- Palmgren
- 2003
(Show Context)
Citation Context ...5. Conclusions Since partially ordered sets are special small categories, presheaf models give as a special case extensions for constructive set theories of Kripke models for intuitionistic logic. In =-=[24]-=- Erik Palmgren applied a Kripke model construction to show an independence result for first-order intuitionistic logic. The result regards the notionsPRESHEAF MODELS FOR CONSTRUCTIVE SET THEORIES 13 o... |