Classifying topoi in synthetic guarded domain theory

TL;DR

This paper demonstrates that various topos models of synthetic guarded domain theory classify simple geometric theories and introduces a modular universal property for multi-clock guarded recursion, unifying complex models.

Contribution

It shows that important topos models of SGDT classify simple theories and formalizes multi-clock guarded recursion as a universal, modular construction applicable to any single-clock topos model.

Findings

Topos models of SGDT classify simple geometric theories.

Multi-clock guarded recursion can be understood via lower bagtopos constructions.

Universal property of multi-clock guarded recursion is modular and broadly applicable.

Abstract

Several different topoi have played an important role in the development and applications of synthetic guarded domain theory (SGDT), a new kind of synthetic domain theory that abstracts the concept of guarded recursion frequently employed in the semantics of programming languages. In order to unify the accounts of guarded recursion and coinduction, several authors have enriched SGDT with multiple "clocks" parameterizing different time-streams, leading to more complex and difficult to understand topos models. Until now these topoi have been understood very concretely qua categories of presheaves, and the logico-geometrical question of what theories these topoi classify has remained open. We show that several important topos models of SGDT classify very simple geometric theories, and that the passage to various forms of multi-clock guarded recursion can be rephrased more compositionally in terms of the lower bagtopos construction of Vickers and variations thereon due to Johnstone. We contribute to the consolidation of SGDT by isolating the universal property of multi-clock guarded recursion as a modular construction that applies to any topos model of single-clock guarded recursion.