The Synthetic Sierpiński Cone

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.

• The paper establishes that within certain reflective subuniverses in HoTT, the Sierpiński cone precisely classifies partial maps, underlining its universal property.
• It rigorously distinguishes between Segal, based Segal, and Sierpiński completeness, demonstrating strict separations with Brouwerian counterexamples.
• The work extends the Sierpiński cone concept to mapping cylinders, unifying geometric insights with type-theoretic localization techniques.

The Synthetic Sierpiński Cone: An Authoritative Overview

Introduction and Context

"The Synthetic Sierpiński Cone" (2605.00773) delivers a comprehensive investigation into the role and structural properties of the Sierpiński cone construction within synthetic topology, synthetic higher category theory, and synthetic domain theory, primarily formalized internally to homotopy type theory (HoTT) with interval objects. The analysis incorporates partial map classifiers, lax colimits, reflective subuniverses, and the classification of open and partial maps, situating these results amid a broad spectrum of contemporary research in synthetic mathematics.

The Sierpiński cone, denoted $X_\bot$ for a space or type $X$, attaches a universal "undefined" or closed point below all others, functioning as a lax colimit akin to the free addition of an initial object in ordinary category theory. Intrinsically linked is the partial map classifier $\Lift(X)$, which classifies open partial maps into $X$. The classical fact that $X_\bot \to \Lift(X)$ is an equivalence is scrutinized in the synthetic setting, where dependent type-theoretic invariance and context-sensitivity render such universal statements too strong and lead to degeneracies unless restricted to carefully chosen subuniverses.

Universal Properties, Partial Map Classification, and Synthetic Foundations

The paper rigorously examines the universal property of the Sierpiński cone in synthetic settings, showing it only robustly classifies partial maps on "global" types or within certain localizations. For arbitrary parameterized types, enforcing the $X_\bot \simeq \Lift(X)$ equivalence universally forces all spaces to become codiscrete, collapsing the theory's expressive power.

Instead, the analysis identifies reflective subuniverses—specifically, accessible localizations at families of "little" Sierpiński comparison maps parameterized by the interval—within which the universal property is nondegenerate and aligns precisely with the partial map classifier characterization. This maximal subuniverse sits strictly inside the Segal types, demarcated by the inner horn orthogonality condition, which arises in synthetic $\infty$-category theory.

Synthetic Notions: Segal, Based Segal, and Sierpiński Completeness

Central to the paper are the notions of Segal, based Segal, Sierpiński, and little Sierpiński completeness for types, defined via left orthogonality to critical maps:

• Segal completeness: orthogonality to the inner horn inclusion $\Horn \hookrightarrow \Spx{2}$.
• Based Segal completeness: orthogonality to the family $(i=1)_\bot \hookrightarrow \II/i$ for $i:\II$ (the "little" cones).
• Sierpiński completeness: orthogonality to all Sierpiński comparison maps $X$0 for $X$1-connected $X$2.
• Little Sierpiński completeness: orthogonality to the family $X$3.

A major result is the strict hierarchy among these classes in all nontrivial distributive lattice contexts satisfying Phoa's principle, particularly that based Segal completeness is strictly stronger than classical Segal completeness. This disambiguates prior conjectures about their coincidence, showing via Brouwerian-style counterexamples that codiscreteness or quotient-initiality would result were they equal.

The equivalence between Sierpiński and little Sierpiński completeness is established, solving an open problem from prior work [pugh-sterling-2025]: a type is Sierpiński complete iff it is little Sierpiński complete. Consequently, Sierpiński complete types form an accessible reflective localization.

Mapping Cylinders and Generalizations

The paper refines the geometric perspective by generalizing the Sierpiński cone to open display and mapping cylinders. The main categorical insight is that the mapping cylinder of a map $X$4 is, in the synthetic setting, the sum of Sierpiński cones on the fibers of $X$5. The analysis shows that within the Sierpiński complete subuniverse, the open mapping cylinder realizes a right-handed universal property, classifying points of $X$6 equipped with open partial elements of the fiber $X$7. This bridges the geometry of mapping cylinders and the logic of partial map classifiers, providing new structural clarity.

Further, the work employs abstract localization technology, leveraging little Sierpiński completeness to show the classes of Segal and Sierpiński complete types are closed under formation of open partial map classifiers and their relative versions. This closure property is essential for the computational and categorical robustness of these subuniverses.

Methodological and Foundational Choices

All results are formalized within univalent foundations (HoTT) with minimal additional axioms, parameterizing statements in a bounded distributive lattice $X$8 rather than assuming a distinguished interval object with strong properties. This approach underpins the generality and mechanizability of the results, ensuring applicability beyond the standard model of simplicial sets to settings derived from domain theory, lattice theory, or Stone duality.

The paper connects these structures to broader themes in the logic–geometry duality, synthetic differential geometry, Stone duality, and synthetic computability. In denotational semantics, Sierpiński completeness characterizes types $X$9 where case analysis on the definedness of lazy functions $\Lift(X)$0 is possible, linking to observable preorders and properties such as path transitivity. In higher category theory, the inability for all synthetic $\Lift(X)$1-categories to be Sierpiński complete implies significant limitations to universal constructions like free initial object addition, demanding context-sensitive approaches.

Strong Numerical and Structural Results

Several key technical achievements include:

• Rigorous proof that Sierpiński completeness for arbitrary types coincides with little Sierpiński completeness, not just on 0-truncated types.
• Demonstration that based Segal completeness and Segal completeness are strictly separated in general, except in degenerate "two-element" interval settings, with supporting counterexamples.
• Closure under (relative) open partial map classifier formation for Segal, based Segal, and Sierpiński complete types under mild assumptions on $\Lift(X)$2 (local, dominant, satisfies Phoa's principle).

Implications and Future Directions

Practically, these results illuminate the design and semantics of type theories and proof assistants seeking synthetic characterizations of categories, domains, or topological spaces, providing a toolkit for constructing well-behaved subuniverses supporting convenient geometric and logical structure. The strict separation between Segal and based Segal conditions, and the identification of the reflective subuniverse where Sierpiński cone universal properties hold, delimit the possibilities for expressing partiality, case analysis, and colimits in such settings.

Theoretically, the work contributes to the understanding of locality in type theory, the granular control of colimit constructions, and the interplay between logical and geometric methods in the synthetic context. The mechanics of the comparision map and reflection processes used here could spur further advances in the explicit computational handling of universal properties in synthetic higher category theory.

Looking forward, the investigation of these structures in models of synthetic geometry, or in the further unified frameworks motivated by Blechschmidt's synthetic quasi-coherence, may yield deeper connections or applications in topos-theoretic algebraic geometry, modality-rich type systems, or semantic models of computation.

Conclusion

This paper solidifies the position of the Sierpiński cone and its universal property as a foundational but subtle tool in synthetic topology and category theory. By precisely characterizing the reflective subuniverses supporting its classifying role, delineating strict containment relations among key completeness notions, and extending the theory to mapping cylinders and localizations, it provides a robust platform for ongoing research at the intersection of categorical logic, synthetic mathematics, and type-theoretic foundations.

Reference:

"The Synthetic Sierpiński Cone" (2605.00773)