Reference. When is the partial map classifier a Sierpiński cone? [sterling-2025-yamcats-37]

The idea of synthetic domain theory is to work in the internal language of a topos containing an interval object that forms a dominance and satisfies a few other axioms, such as Phoa’s principle. Then, not all objects deserve to be called “predomains”, but the ones that do invariably arise within full internal reflective subcategories defined by simple orthogonality laws stated in terms of the interval’s geometry. It so happens that some of these orthogonality laws are also of use in defining synthetic ∞-categories à la Riehl and Shulman.

I will outline some recent results in synthetic (higher) domain theory obtained with my former Masters student Leoni Pugh concerning partial map classifiers, including (1) the closure of synthetic ∞-categories under partial map classifiers, and (2) the discovery of a strengthening of the Segal completeness law that, amongst synthetic partial orders, causes the partial map classifier to coincide with the Sierpiński cone.

(See our joint paper When is the partial map classifier a Sierpiński cone?, to appear in LICS ’25: 40th Annual ACM/IEEE Symposium on Logic in Computer Science.)