The Synthetic Sierpiński Cone
For researchers in synthetic higher category theory and domain theory, this clarifies the limitations and scope of the Sierpiński cone's universal property in synthetic settings.
The paper studies the Sierpiński cone in synthetic homotopy type theory, showing that while it classifies partial maps globally, this property fails for parameterised types. It identifies the largest subuniverse where classification holds as an accessible localisation, contained within Segal types, and extends results to mapping cylinders.
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.