Domains and Classifying Topoi
Jonathan Sterling, Lingyuan Ye

TL;DR
This paper establishes a novel connection between synthetic domain theory and Grothendieck topoi, revealing new techniques and models for reasoning about domain-like structures in a categorical framework.
Contribution
It introduces a countable synthetic quasi-coherence principle that unifies various synthetic theories and provides new dualities and models for synthetic domain theory.
Findings
Derived all axioms of synthetic domain theory from the quasi-coherence principle
Established a duality between quasi-coherent algebras and affine spaces in a topos
Identified a broad class of sheaf models for synthetic domain theory
Abstract
We explore a new connection between synthetic domain theory and Grothendieck topoi related to the distributive lattice classifier. In particular, all the axioms of synthetic domain theory (including the inductive fixed point object and the chain completeness of the dominance) emanate from a countable version of the synthetic quasi-coherence principle that has emerged as a central feature in the unification of synthetic algebraic geometry, synthetic Stone duality, and synthetic category theory. The duality between quasi-coherent algebras and affine spaces in a topos with a distributive lattice object provides a new set of techniques for reasoning synthetically about domain-like structures, and reveals a broad class of (higher) sheaf models for synthetic domain theory.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Logic, programming, and type systems · Topological and Geometric Data Analysis
