## Sheaf Toposes for Realizability (2001)

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### BibTeX

@TECHREPORT{Awodey01sheaftoposes,

author = {Steven Awodey and Andrej Bauer},

title = {Sheaf Toposes for Realizability},

institution = {},

year = {2001}

}

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### Abstract

We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over di#erent partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott. 1

### Citations

837 |
Theory of recursive functions and effective computability
- Rogers
- 1967
(Show Context)
Citation Context ...s the set of natural numbers N equipped with Kleene application n · m = {n}m which applies the n-th partial recursive function {n} to m. The existence of K and S is a consequence of the s-m-n theorem =-=[Rog87]-=-. Example 2.4 The second Kleene Algebra B = N N , or the Baire space, is a partial combinatory algebra. The continuous partial application is defined as follows. If α ∈ N N , let αn be the sequence [α... |

231 |
Computable Analysis
- Weihrauch
- 2000
(Show Context)
Citation Context ...eserving continuous maps. Two such maps are considered equivalent when they map equivalent points to equivalent points. Example 2.8 The category Mod(B) is the ambient category of Type Two Effectivity =-=[Wei00]-=-. It is equivalent to 0-equilogical spaces [Bau00], which is the full subcategory on those countably based equilogical spaces whose underlying topological space is 0-dimensional. Any discrete applicat... |

210 |
Data types as lattices
- SCOTT
- 1976
(Show Context)
Citation Context ...er and they are equal. We usually write xy instead of x · y, and associate application to the left. Here we only consider non-trivial PCAs that satisfy K �= S. Example 2.2 Scott’s graph model P ==-= PN, [Sco76], is-=- a model of the untyped λ-calculus, therefore a (total) combinatory algebra. The algebraic lattice of continuous maps P P is a retract of the algebraic lattice P. The embedding Γ: P P → P is defin... |

188 |
Sheaves in Geometry and Logic. A First Introduction to Topos Theory
- Lane, Moerdijk
- 1992
(Show Context)
Citation Context ...], where the focus was mainly on the locally cartesian closed structure of realizability models. Here we are also interested in comparison of logical properties. As a reference on topos theory we use =-=[MM92], -=-and Johnstone and Moerdijk [JM89] as a reference on local map of toposes. Longley’s dissertation [Lon94] contains material on partial combinatory algebras, applicative morphisms, and realizability. ... |

60 |
Realizability Toposes and Language Semantics
- Longley
- 1994
(Show Context)
Citation Context ...are also interested in comparison of logical properties. As a reference on topos theory we use [MM92], and Johnstone and Moerdijk [JM89] as a reference on local map of toposes. Longley’s dissertatio=-=n [Lon94] con-=-tains material on partial combinatory algebras, applicative morphisms, and realizability. In Birkedal’s dissertation [Bir00] and in ∗ 5000 Forbes Ave., Pittsburgh, PA 15213, USA; E-mail: awodey@an... |

45 |
Continuity and Effectiveness in Topoi
- Rosolini
- 1986
(Show Context)
Citation Context ...algebras. This work is related to Rosolini and Streicher [RS99], where the focus was mainly on the locally cartesian closed structure of realizability models, as well as to Mulry [Mul82] and Rosolini =-=[Ros86]-=-. Here we are also interested in comparison of logical properties. As a reference on topos theory we use [MM92], and Johnstone and Moerdijk [JM89] as a reference on local map of toposes. Longley’s dis... |

42 |
Countable functionals
- Kleene
- 1959
(Show Context)
Citation Context ...lving NNN is valid in Mod(B) but not in Mod(P): ∀ F ∈ N NN . ∃ α ∈ N N . ∀ β ∈ N N . F β = (α|β) . (2) The sentence states that every functional F ∈ NNN has an associate α ∈ NN i=-=n the sense of Kleene [Kle59]. Here α|β is Kleene’s conti-=-nuous function application. The statement n = (α|β) is equivalent to ∃ m ∈ N . � α(βm) = n + 1 ∧ ∀ k ∈ N . � k < m −→ α(βk) = 0 �� . Similarly, a statement that all functio... |

41 | The Realizability Approach to Computable Analysis and Topology
- Bauer
- 2000
(Show Context)
Citation Context ...ered equivalent when they map equivalent points to equivalent points. Example 2.8 The category Mod(B) is the ambient category of Type Two Effectivity [Wei00]. It is equivalent to 0-equilogical spaces =-=[Bau00], which is t-=-he full subcategory on those countably based equilogical spaces whose underlying topological space is 0-dimensional. Any discrete applicative morphism ρ: E PCA −−−→ F induces a functor �ρ:... |

37 |
The- Foundations of Intuitionistic= Mathematics, especially in relation to recursive functions. Studies in Logic and the
- Kleene, Vesley
- 1965
(Show Context)
Citation Context ... | β)n = α ⋆ (n::β) , where n::β is the sequence obtained by prepending n to β. We consider α |β to be defined only if (α | β)n is defined for all n ∈ N. For details about the PCA structu=-=re of B, see [KV65]. Definition 2.4 (J-=-ohn Longley) Let E and F be PCAs. An applicative morphism ρ: E PCA −−−→ F is a total relation ρ ⊆ E × F for which there exists r ∈ F, such that rx ↓ for all x ∈ F, and for all u, v ... |

31 | Equilogical spaces
- Bauer, Birkedal, et al.
(Show Context)
Citation Context ...A) is equivalent to the perhaps better known category PER(A) of partial equivalence relations on A. Example 2.7 The category Mod(P) is equivalent to the category of countably based equilogical spaces =-=[BBS98]. A counta-=-bly based equilogical space is a pair (X, ≡X) where X is a countably based T0-space and ≡ is an equivalence relation on X. A morphism [f]: (X, ≡X) → (Y, ≡Y ) is an equivalence class of equiv... |

29 |
Proper maps of toposes
- Moerdijk, Vermeulen
(Show Context)
Citation Context ...e locally cartesian closed structure of realizability models. Here we are also interested in comparison of logical properties. As a reference on topos theory we use [MM92], and Johnstone and Moerdijk =-=[JM89] as -=-a reference on local map of toposes. Longley’s dissertation [Lon94] contains material on partial combinatory algebras, applicative morphisms, and realizability. In Birkedal’s dissertation [Bir00] ... |

24 | Local realizability toposes and a modal logic for computability
- Awodey, Birkedal, et al.
(Show Context)
Citation Context .... In Birkedal’s dissertation [Bir00] and in ∗ 5000 Forbes Ave., Pittsburgh, PA 15213, USA; E-mail: awodey@andrew.cmu.edu † Auravägen 17, SE-182 62 Djursholm, Sweden; E-mail: Andrej.Bauer@andrej=-=.com 1s[ABS99] you-=- can find further information about realizability, and also on local maps of toposes and the corresponding ♯-♭ calculus. 2 Sheaves on Partial Combinatory Algebras We quickly review the basic defin... |

20 | Developing theories of types and computability via realizability
- Birkedal
- 2000
(Show Context)
Citation Context ...k [JM89] as a reference on local map of toposes. Longley’s dissertation [Lon94] contains material on partial combinatory algebras, applicative morphisms, and realizability. In Birkedal’s dissertat=-=ion [Bir00] and -=-in ∗ 5000 Forbes Ave., Pittsburgh, PA 15213, USA; E-mail: awodey@andrew.cmu.edu † Auravägen 17, SE-182 62 Djursholm, Sweden; E-mail: Andrej.Bauer@andrej.com 1s[ABS99] you can find further informa... |

14 |
Generalized Banach-Mazur functionals in the topos of recursive sets
- Mulry
- 1982
(Show Context)
Citation Context ... partial combinatory algebras. This work is related to Rosolini and Streicher [RS99], where the focus was mainly on the locally cartesian closed structure of realizability models, as well as to Mulry =-=[Mul82]-=- and Rosolini [Ros86]. Here we are also interested in comparison of logical properties. As a reference on topos theory we use [MM92], and Johnstone and Moerdijk [JM89] as a reference on local map of t... |

11 | Introduction to Set Theory. Second edition - Hrbacek, Jech - 1984 |

9 | Developing Theories of Types and Computability
- Birkedal
- 1999
(Show Context)
Citation Context ...δ∗ �� Sh(F) where δ ∗ ◦ η ∗ ∼ = 1 Sh(F). Thus, a +-preserving discrete applicative retraction (η ⊣ δ): E PCA −−−→ F induces a local map of toposes Sh(E) → Sh(F). Such ma=-=ps were studied by Birkedal in [Bir99]; th-=-ere and in [ABS99] a ♯-♭ calculus for the internal logic is obtained, which can be used here to compare realizability in modest sets over E with that over F. An applicative retraction does not see... |

3 |
Comparing models of higher type computation
- Streicher
- 1999
(Show Context)
Citation Context ...nery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over different partial combinatory algebras. This work is related to Rosolini and Streicher =-=[RS99]-=-, where the focus was mainly on the locally cartesian closed structure of realizability models. Here we are also interested in comparison of logical properties. As a reference on topos theory we use [... |