Tensorial structure of the lifting doctrine in constructive domain theory

2023-12-28

arXiv.org

AI Summary

This paper systematically investigates the structure and properties of lift operators in constructive domain theory. Employing category theory, 2-category theory, topos semantics, and constructive mathematics, it constructs—*for the first time in an arbitrary elementary topos*—the 2-categorical and tensor structures of lift operators and establishes their universal property as the Sierpiński cone. Key contributions are: (1) proving that the lift operator is a Kock–Zöberlein monad, removing classical non-constructive assumptions; (2) establishing a 2-categorical equivalence among lift algebras, pointed dcpos, and inductive partial orders; (3) demonstrating that the category of lift algebras is cocomplete, symmetric monoidal closed, and that connected colimits are created by the forgetful functor; and (4) showing that the smash product, within the constructive setting, classifies both bilinear and strict maps, with explicit computations of several coequalizers.

Abstract

We present a survey of the two-dimensional and tensorial structure of the lifting doctrine in constructive domain theory, i.e. in the theory of directed-complete partial orders (dcpos) over an arbitrary elementary topos. We establish the universal property of lifting of dcpos as the Sierpi'nski cone, from which we deduce (1) that lifting forms a Kock-Z""oberlein doctrine, (2) that lifting algebras, pointed dcpos, and inductive partial orders form canonically equivalent locally posetal 2-categories, and (3) that the category of lifting algebras is cocomplete, with connected colimits created by the forgetful functor to dcpos. Finally we deduce the symmetric monoidal closure of the Eilenberg-Moore resolution of the lifting 2-monad by means of smash products; these are shown to classify both bilinear maps and strict maps, which we prove to coincide in the constructive setting. We provide several concrete computations of the smash product as dcpo coequalisers and lifting algebra coequalisers, and compare these with the more abstract results of Seal. Although all these results are well-known classically, the existing proofs do not apply in a constructive setting; indeed, the classical analysis of the Eilenberg-Moore category of the lifting monad relies on the fact that all lifting algebras are free, a condition that is not known to hold constructively.

Problem

• Domain Theory
• Bilifting Algebra
• Kock-Zoberlein Monads

Innovation

• Sierpinski cones
• Kock-Zöberlein rule
• smash product