The Constructive Geometry of the Lifting Doctrine in Domain Theory

Conceitos essenciais

The lifting monad on directed-complete partial orders (dcpos) has a rich 2-categorical structure, including universal properties as a Sierpiński cone and a partial product. This enables a constructive analysis of the lifting doctrine, including the cocompleteness of lifting algebras and their symmetric monoidal closed structure.

Resumo

The paper presents a comprehensive analysis of the tensorial structure of the lifting doctrine in constructive domain theory, i.e., the theory of directed-complete partial orders (dcpos) over an arbitrary elementary topos. The key insights are:

The universal property of lifting of dcpos as the Sierpiński cone is established, from which it is deduced that (1) lifting forms a Kock–Zöberlein doctrine, (2) lifting algebras, pointed dcpos, and inductive partial orders form canonically equivalent locally posetal 2-categories, and (3) the category of lifting algebras is cocomplete, with connected colimits created by the forgetful functor to dcpos.

The symmetric monoidal closure of the Eilenberg–Moore resolution of the lifting 2-monad is deduced by means of smash products, which are shown to classify both bilinear maps and strict maps, which are proven to coincide in the constructive setting. Several concrete computations of the smash product as dcpo and lifting algebra coequalisers are provided.

The paper highlights the importance of studying constructive domain theory, as modern approaches to programming semantics routinely involve computing recursive functions in non-boolean toposes, where the constructive theory of dcpos is necessary. Many results that appear "obvious" classically have not been established constructively, and the constructive domains behave differently enough from the classical ones that it would not be safe to take these results for granted.

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Principais Insights Extraídos De

Tensorial structure of the lifting doctrine in constructive domain theory

by Jonathan Ste... às arxiv.org 04-17-2024

https://arxiv.org/pdf/2312.17023.pdf

Perguntas Mais Profundas

How can the results on the cocompleteness and symmetric monoidal closed structure of lifting algebras be extended to the setting of locales and toposes

The results on the cocompleteness and symmetric monoidal closed structure of lifting algebras in the context of dcpos can potentially be extended to locales and toposes. In locales, the concept of cocompleteness is well-defined, and the construction of colimits can be adapted to the locale setting. The key would be to establish the appropriate universal properties and categorical structures that mirror those in the constructive domain theory of dcpos. By understanding how colimits and symmetric monoidal structures work in locales, we can potentially generalize the results obtained for lifting algebras in dcpos to the setting of locales.
When it comes to toposes, the situation is more complex due to the richer structure and logic involved. Topos theory provides a framework for understanding constructive mathematics in a broader context. Extending the results on lifting algebras to toposes would require a deep dive into the categorical properties of toposes, such as internal logic, subobject classifiers, and geometric morphisms. By leveraging the foundational concepts of topos theory, it may be possible to establish analogous results for lifting algebras in the context of toposes.

Are there examples of non-free lifting algebras in constructive mathematics, or can all lifting algebras be shown to be free on their subdcpo of positive elements

In constructive mathematics, the question of whether all lifting algebras are free on their subdcpo of positive elements is a significant one. While classically every lifting algebra is free, this condition does not necessarily hold constructively. The existence of non-free lifting algebras in constructive mathematics would imply a departure from the classical understanding of lifting algebras.
To determine if there are examples of non-free lifting algebras in constructive mathematics, one would need to carefully analyze the properties and structures of lifting algebras in a constructive setting. This analysis would involve investigating the relationships between lifting algebras and their positive elements, as well as exploring the universal properties and categorical structures that define lifting algebras. By delving into the specific characteristics of lifting algebras in constructive domain theory, one can potentially identify instances where lifting algebras are not free on their positive elements.

What are the connections between the 2-categorical universal properties of the lifting doctrine and the effective and variable set-theoretic approaches to programming semantics

The 2-categorical universal properties of the lifting doctrine play a crucial role in understanding the effective and variable set-theoretic approaches to programming semantics. These universal properties provide a formal framework for reasoning about lifting algebras, partial maps, and other constructs in a constructive domain theory. By establishing the connections between the lifting doctrine and programming semantics, we can enhance our understanding of how computational processes can be modeled and analyzed in a constructive setting.
Effective and variable set-theoretic approaches in programming semantics often involve dealing with partial functions, nondeterministic computations, and varying data structures. The lifting doctrine, with its focus on lifting algebras and partial maps, offers a way to formalize and reason about these computational aspects. By leveraging the 2-categorical universal properties of the lifting doctrine, we can develop a systematic and rigorous framework for studying the interactions between constructive domain theory and programming semantics. This framework enables us to address key computational challenges such as handling partiality, nondeterminism, and dynamic data structures in a constructive and mathematically sound manner.