When is the partial map classifier a Sierpiński cone?

Abstract

We study the relationship between partial map classifiers, Sierpiński cones, and axioms for synthetic higher categories and domains within univalent foundations. In particular, we show that synthetic $\infty$-categories are closed under partial map classifiers assuming Phoa's principle, and we isolate a new reflective subuniverse of types within which the Sierpiński cone (a lax colimit) can be computed as a partial map classifier by strengthening the Segal condition.

To appear in to LICS 2025. Main result updated to omit unnecessary side condition (boundary separation)