## Partial Combinatory Algebras and Realizability Toposes (2004)

### BibTeX

@MISC{Hofstra04partialcombinatory,

author = {Pieter J. W. Hofstra},

title = {Partial Combinatory Algebras and Realizability Toposes},

year = {2004}

}

### OpenURL

### Abstract

These are the lecture notes for a tutorial at FMCS 2004 in Kananaskis. The aim is to give a first introduction to Partial Combinatory Algebras and the construction of Realizability Toposes. The first part, where Partial Combinatory Algebras are discussed, requires no specific background (except for some of the examples perhaps), although familiarity with combinatory logic and lambda calculus will not hurt. The second part on realizability toposes presupposes some knowledge of category theory; more specifically, we will assume that the reader knows what a topos is. Apart from that the material is self-contained. 1 Partial Combinatory Algebras We give the basic definitions and properties of Partial Combinatory Algebras in the first subsection. Next, we discuss some of the important examples. Finally, we touch upon the theory of Partial Combinatory Algebras. 1.1 Partial Applicative Structures and Combinatory Completeness We first introduce the basic concept of a Partial Applicative Structure, which may be viewed as a universe for computation. Then look at terms over an applicative structure, we formulate

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(Show Context)
Citation Context ...btains by taking D to be a complete atomic Boolean algebra (i.e. a powerset). It can be shown that for total combinatory algebras one has the following proper inclusions For more on this, we refer to =-=[2, 1]-=-. CAs ⊂ λ-algebras ⊂ λ-models ⊂ extensional CAs. Totality and Completability. We already saw that a PCA A is called total if ab ↓ for all a, b ∈ A. Scott’s Pω, for example, is total, whereas Kleene’s ... |

87 |
The effective topos
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(Show Context)
Citation Context ...s out with Kleene’s PCA K1, then the resulting topos is known as the Effective Topos, Eff. This was the first realizability topos that was investigated, and its discovery is due to Martin Hyland (see =-=[4]-=-). So far, we have not explained where the nomenclature “realizability topos“ comes from. Realizability is an interpretation for a formal system of intuitionistic arithmetic called Heyting Arithmetic.... |

87 |
On the Interpretation of Intuitionistic Number Theory
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Citation Context ...ealizability topos“ comes from. Realizability is an interpretation for a formal system of intuitionistic arithmetic called Heyting Arithmetic. This interpretation, devised by S.C. Kleene in 1945 (see =-=[7, 10]-=-), is based on recursion theory and has had many applications in proof theory and metamathematics of constructive systems. Since the PCA K1 is also based on recursive functions one can expect a relati... |

39 |
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Citation Context ...a combinatory algebra structure. Other Examples. There are other kinds of PCAs which we will not go into here. Just to mention a few: Kleene’s K2 has the set of functions N → N as underlying set (see =-=[8]-=-). Also, one can consider generalizations of PCAs, such as ordered PCAs. We will briefly return to these when discussing realizability toposes. Finally, one has term models, obtained by taking equival... |

29 |
The theory of triposes
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(Show Context)
Citation Context ...BCC and has a generic predicate is called a tripos. This is an acronym for Topos Representing Indexed PreOrdered Set. The precise definition is a bit more subtle than what we’ve seen, and we refer to =-=[9, 5]-=- for details and many more examples of triposes. The intuition that should be kept in mind is that a tripos is a setting in which one can interpret intuitionistic, higher-order typed logic without equ... |

26 | The discrete objects in the effective topos
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(Show Context)
Citation Context ...ovide a model for constructive analysis. Finally, there is a famous result stating that there exists an internal category in Eff which is small, complete, but not a poset. For a detailed account, see =-=[6]-=-. For ordinary categories, this is impossible by a theorem by Peter Freyd. This category is the internalization of a full subcategory of Eff, called P ER (Partial Equivalence Relations), which plays a... |

15 | Collapsing partial combinatory algebras
- Bethke, Klop
- 1996
(Show Context)
Citation Context ...phic to D∞. As for Graph Algebras, these are not extensional but can be made extensional by quotienting out by a suitable equivalence relation. Such a procedure is known as “extensional collapse”. In =-=[2]-=- it is shown that these structures are a special instance of the D∞ models. In fact, graph algebras, made extensional, are precisely those D∞ that one obtains by taking D to be a complete atomic Boole... |

11 | Oosten. Ordered partial combinatory algebras
- Hofstra, van
- 2003
(Show Context)
Citation Context ...morphism. Given the construction of a topos RT(A) from a PCA A, one might wonder which homomorphisms of PCAs induce geometric morphisms between the realizability toposes. This problem is addressed in =-=[3]-=-. It turns out that a very different notion of homomorphism is needed than the “model-theoretic” homomorphism. First, we consider a generalization of PCAs, which we call Ordered PCAs. This is a poset ... |