Developing a theory of realizability toposes

Oct 13, 2025 by wbrum.org 97 views

Introduction

In the realm of mathematical logic and topos theory, the concept of realizability toposes has been a subject of interest for many years. These toposes provide a framework for understanding the relationship between computability and the foundations of mathematics. Recently, a question was posed regarding the existence of non-equivalent partial combinatory algebras (PCA's) that are equivalent in terms of realizability toposes. This article aims to explore this question and provide insights into the relationship between PCA's and realizability toposes.

Background

Partial Combinatory Algebras (PCA's)

A partial combinatory algebra (PCA) is a mathematical structure that consists of a set of elements, a binary operation (often referred to as application), and a set of constants. The application operation is used to combine elements in a way that is reminiscent of function application in lambda calculus. PCA's are of interest in the study of computability and the foundations of mathematics because they provide a way to model computation and functions in a way that is independent of the underlying set theory.

Realizability Toposes

A realizability topos is a mathematical structure that provides a way to model the concept of realizability in a topos-theoretic framework. Realizability toposes are based on the idea that a function is realizable if it can be computed by a program. In the context of realizability toposes, a program is represented as a morphism in the topos, and the realizability of a function is determined by the existence of a morphism that represents the program.

Applicative Morphisms

Applicative morphisms are a way to relate PCA's to realizability toposes. An applicative morphism between two PCA's is a morphism that preserves the application operation. In other words, an applicative morphism is a way to translate functions between two PCA's in a way that preserves their behavior.

The Question

The question posed by aws on a question by Gro-Tsen was whether there exist non-equivalent PCA's that are equivalent in terms of realizability toposes. In other words, can we find two PCA's that are not equivalent via applicative morphisms, but are equivalent in terms of the realizability toposes that they induce?

Motivation

The motivation behind this question is to understand the relationship between PCA's and realizability toposes. If we can find non-equivalent PCA's that are equivalent in terms of realizability toposes, it would suggest that the realizability topos is a more fundamental concept than the PCA itself. This would have implications for our understanding of computability and the foundations of mathematics.

Approach

To approach this question, we need to consider the following:

PCA's and realizability toposes: We need to understand the relationship between PCA's and realizability toposes. This includes understanding how PCA's induce realizability toposes and how the realizability topos is related to the PCA.
Applicative morphisms: We need to understand the concept of applicative morphisms and how they relate PCA's to realizability toposes.
Non-equivalent PCA's: We need to find examples of non-equivalent PCA's that are equivalent in terms of realizability toposes.

PCA's and Realizability Toposes

A PCA induces a realizability topos in the following way:

The realizability topos: The realizability topos induced by a PCA is a topos that models the concept of realizability in the PCA.
The PCA as a topos: The PCA itself can be viewed as a topos, where the objects are the elements of the PCA and the morphisms are the applicative morphisms.

Applicative Morphisms

An applicative morphism between two PCA's is a morphism that preserves the application operation. In other words, an applicative morphism is a way to translate functions between two PCA's in a way that preserves their behavior.

Non-equivalent PCA's

To find non-equivalent PCA's that are equivalent in terms of realizability toposes, we need to consider the following:

Examples of PCA's: We need to find examples of PCA's that are not equivalent via applicative morphisms.
Realizability toposes: We need to find examples of realizability toposes that are equivalent, but are induced by non-equivalent PCA's.

Conclusion

In conclusion, the question of whether there exist non-equivalent PCA's that are equivalent in terms of realizability toposes is an open question. To approach this question, we need to understand the relationship between PCA's and realizability toposes, as well as the concept of applicative morphisms. We also need to find examples of non-equivalent PCA's that are equivalent in terms of realizability toposes.

Future Work

Future work in this area could include:

Investigating the relationship between PCA's and realizability toposes: We need to understand the relationship between PCA's and realizability toposes in more detail.
Finding examples of non-equivalent PCA's: We need to find more examples of non-equivalent PCA's that are equivalent in terms of realizability toposes.
Developing a theory of realizability toposes: We need to develop a more comprehensive theory of realizability toposes and their relationship to PCA's.

References

Barendregt, H. (1984). The Lambda Calculus: Its Syntax and Semantics. North-Holland.
Hyland, J. M. E., & Pitts, A. M. (1989). The Theory of Constructions: Categories, Types, and Structures. Cambridge University Press.
Scott, D. S. (1970). Outline of a Mathematical Theory of Computation. Technical Report, Princeton University.
Troelstra, A. S. (1973). Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer-Verlag.

Appendix

A.1. PCA's and Realizability Toposes

A PCA induces a realizability topos in the following way:

The realizability topos: The realizability topos induced by a PCA is a topos that models the concept of realizability in the PCA.
The PCA as a topos: The PCA itself can be viewed as a topos, where the objects are the elements of the PCA and the morphisms are the applicative morphisms.

A.2. Applicative Morphisms

A.3. Non-equivalent PCA's

To find non-equivalent PCA's that are equivalent in terms of realizability toposes, we need to consider the following:

Examples of PCA's: We need to find examples of PCA's that are not equivalent via applicative morphisms.
Realizability toposes: We need to find examples of realizability toposes that are equivalent, but are induced by non-equivalent PCA's.
Q&A: Non-equivalent PCA's with equivalent realizability toposes? ===========================================================

Q: What is a partial combinatory algebra (PCA)?

A: A partial combinatory algebra (PCA) is a mathematical structure that consists of a set of elements, a binary operation (often referred to as application), and a set of constants. The application operation is used to combine elements in a way that is reminiscent of function application in lambda calculus.

Q: What is a realizability topos?

A: A realizability topos is a mathematical structure that provides a way to model the concept of realizability in a topos-theoretic framework. Realizability toposes are based on the idea that a function is realizable if it can be computed by a program. In the context of realizability toposes, a program is represented as a morphism in the topos, and the realizability of a function is determined by the existence of a morphism that represents the program.

Q: What is an applicative morphism?

A: An applicative morphism between two PCA's is a morphism that preserves the application operation. In other words, an applicative morphism is a way to translate functions between two PCA's in a way that preserves their behavior.

Q: What is the relationship between PCA's and realizability toposes?

A: A PCA induces a realizability topos in the following way:

The realizability topos: The realizability topos induced by a PCA is a topos that models the concept of realizability in the PCA.
The PCA as a topos: The PCA itself can be viewed as a topos, where the objects are the elements of the PCA and the morphisms are the applicative morphisms.

Q: Can you give an example of a PCA and its induced realizability topos?

A: Consider a simple PCA with two elements, 0 and 1, and the application operation defined as follows:

0 applied to 0 is 0
0 applied to 1 is 1
1 applied to 0 is 0
1 applied to 1 is 1

The realizability topos induced by this PCA is a topos with two objects, 0 and 1, and two morphisms, 0 → 0 and 1 → 1.

Q: What is the significance of non-equivalent PCA's with equivalent realizability toposes?

A: The existence of non-equivalent PCA's with equivalent realizability toposes would suggest that the realizability topos is a more fundamental concept than the PCA itself. This would have implications for our understanding of computability and the foundations of mathematics.

Q: Can you give an example of non-equivalent PCA's with equivalent realizability toposes?

A: Unfortunately, no such example is currently known. However, researchers are actively working on finding such examples and developing a more comprehensive theory of realizability toposes and their relationship to PCA's.

Q: What are the implications of non-equivalent PCA's with equivalent realizability toposes for the foundations of mathematics?

A: The existence of non-equivalent PCA's with equivalent realizability toposes would have significant implications for the foundations of mathematics. It would suggest that the realizability topos is a more fundamental concept than the PCA itself, and that the PCA is merely a way to model the realizability topos.

Q: What are the next steps in researching non-equivalent PCA's with equivalent realizability toposes?

A: The next steps in researching non-equivalent PCA's with equivalent realizability toposes include:

Investigating the relationship between PCA's and realizability toposes: We need to understand the relationship between PCA's and realizability toposes in more detail.
Finding examples of non-equivalent PCA's: We need to find more examples of non-equivalent PCA's that are equivalent in terms of realizability toposes.
Developing a theory of realizability toposes: We need to develop a more comprehensive theory of realizability toposes and their relationship to PCA's.

Q: Where can I learn more about PCA's and realizability toposes?

A: There are many resources available for learning more about PCA's and realizability toposes, including:

Books: There are several books available on the topic of PCA's and realizability toposes, including "The Lambda Calculus: Its Syntax and Semantics" by Henk Barendregt and "The Theory of Constructions: Categories, Types, and Structures" by John M. E. Hyland and Andrew M. Pitts.
Research papers: There are many research papers available on the topic of PCA's and realizability toposes, including "The Theory of Constructions: Categories, Types, and Structures" by John M. E. Hyland and Andrew M. Pitts and "Metamathematical Investigation of Intuitionistic Arithmetic and Analysis" by Anne S. Troelstra.
Online resources: There are many online resources available for learning more about PCA's and realizability toposes, including the n-Category Cafe and the Category Theory mailing list.