The Synthetic Sierpiński Cone

2026-05-01

AI Summary

This work investigates the capacity of the Sierpiński cone to classify partial maps in synthetic homotopy type theory. We show that requiring it to preserve a right adjoint universal property across all parametrized types leads to degeneracy of the theory. To circumvent this, we identify a maximal reflective subuniverse wherein the Sierpiński cone does classify partial maps, and prove that this subuniverse is strictly contained within the Segal types. This subuniverse admits a characterization as an accessible localization with respect to a family of interval-parametrized embeddings, and when the interval is nontrivial, not all Segal types lie within it. Furthermore, we generalize these results to the mapping cylinder construction, endowing it with a novel right adjoint universal property.

Abstract

In domains, categories, and toposes, the Sierpiński cone construction glues onto a space a universal closed point lying below all the other points. Although this is a lax colimit, it also enjoys a well-known right-handed universal property: the Sierpiński cone classifies partial maps defined on an open subspace. The situation proves more subtle in synthetic models of space based on extending homotopy type theory with an interval, as in several recent approaches to synthetic higher categories and domains: although globally it may well be the case that the Sierpiński cone classifies partial maps, this property cannot hold of all parameterised types without degenerating the theory. On the other hand, there are reflective subuniverses within which the classifying property nonetheless holds. We show that the largest subuniverse in which the Sierpiński cone classifies partial maps is the accessible localisation at a family of embeddings parameterised in the interval, and this subuniverse is contained within the Segal types; this containment is moreover strict in the sense that when the interval is non-trivial, it is not possible for all Segal types to lie in the subuniverse. We finally extend these results from Sierpiński cones to mapping cylinders, providing a new right-handed universal property for the latter.

Problem

• Sierpiński cone
• synthetic spaces
• homotopy type theory
• partial maps
• Segal types

Innovation

• Sierpiński cone
• synthetic topology
• homotopy type theory
• reflective subuniverse
• mapping cylinder