Domain Theory in Constructive and Predicative Univalent Foundations

TL;DR

This paper develops domain theory within constructive univalent foundations, constructing models of PCF and the untyped lambda calculus without resizing axioms, and addresses size issues using universes and accessible categories.

Contribution

It extends domain theory in univalent foundations by constructing Scott models without resizing axioms, handling size issues via universes and accessible categories.

Findings

Constructed Scott models of PCF and lambda calculus in univalent foundations.

Addressed size issues by working with type universes and accessible categories.

Demonstrated that large dcpos can be handled within a predicative framework.

Abstract

We develop domain theory in constructive univalent foundations without Voevodsky's resizing axioms. In previous work in this direction, we constructed the Scott model of PCF and proved its computational adequacy, based on directed complete posets (dcpos). Here we further consider algebraic and continuous dcpos, and construct Scott's $D_\infty$ model of the untyped $\lambda$-calculus. A common approach to deal with size issues in a predicative foundation is to work with information systems or abstract bases or formal topologies rather than dcpos, and approximable relations rather than Scott continuous functions. Here we instead accept that dcpos may be large and work with type universes to account for this. For instance, in the Scott model of PCF, the dcpos have carriers in the second universe $\mathcal{U}_1$ and suprema of directed families with indexing type in the first universe $\mathcal{U}_0$. Seeing a poset as a category in the usual way, we can say that these dcpos are large, but locally small, and have small filtered colimits. In the case of algebraic dcpos, in order to deal with size issues, we proceed mimicking the definition of accessible category. With such a definition, our construction of Scott's $D_\infty$ again gives a large, locally small, algebraic dcpo with small directed suprema.