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arxiv: 2606.29373 · v1 · pith:GGRCZFU7new · submitted 2026-06-28 · 🧮 math.AT · math.CT

An Oriented Street--Roberts Conjecture

David Gepner , Hadrian Heine This is my paper

Pith reviewed 2026-06-30 02:06 UTC · model grok-4.3

classification 🧮 math.AT math.CT
keywords oriented polytopesStreet-Roberts conjecture(∞,∞)-categoriessheaveshigher category theoryGray tensorjoinbicone
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The pith

(∞,∞)-categories arise as sheaves on families of oriented polytopes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines oriented polytopes that include Street's oriented simplices and Gray's oriented cubes. It proves that (∞,∞)-categories correspond exactly to sheaves on certain families of these polytopes. This gives a geometric model in which higher categories appear as directed analogues of homotopy types. The construction induces standard operations such as the join and Gray tensor directly from the geometry of the polytopes and yields formulae for their interactions.

Core claim

We formulate a notion of oriented polytope, including Street's oriented simplices and Gray's oriented cubes, and use this to prove an oriented version of the Street--Roberts conjecture, presenting (∞,∞)-categories as sheaves on suitable families of oriented polytopes, generalizing work of Campion. This allows us to understand (∞,∞)-categories from a geometric perspective, as directed analogues of homotopy types.

What carries the argument

The sheaf condition on families of oriented polytopes, where the polytopes carry orientations that generalize simplices and cubes.

Load-bearing premise

The specific families of oriented polytopes are chosen so that the sheaf condition on them recovers precisely the (∞,∞)-categories.

What would settle it

An explicit presheaf on the chosen families of oriented polytopes that satisfies the sheaf condition yet fails to define an (∞,∞)-category, or a known (∞,∞)-category that fails the sheaf condition on those families.

read the original abstract

We formulate a notion of oriented polytope, including Street's oriented simplices and Gray's oriented cubes, and use this to prove an oriented version of the Street--Roberts conjecture, presenting $(\infty,\infty)$-categories as sheaves on suitable families of oriented polytopes, generalizing work of Campion. This allows us to understand $(\infty, \infty)$-categories from a geometric perspective, as directed analogues of homotopy types. These familes of oriented polytopes induce basic operations in higher category theory: for instance, the join, Gray tensor, and bicone arise from the geometry of the orientals, cubes, and orthoplexes, respectively. We study the interaction of these operations and derive some geometric formulae, generalizing work of Ara--Maltsiniotis, Verity, and others.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript formulates a notion of oriented polytope that includes Street's oriented simplices and Gray's oriented cubes. It proves an oriented version of the Street--Roberts conjecture by exhibiting (∞,∞)-categories as sheaves on suitable families of these polytopes. The result generalizes Campion's work and interprets (∞,∞)-categories geometrically as directed analogues of homotopy types. The families induce operations such as the join, Gray tensor, and bicone from the geometry of orientals, cubes, and orthoplexes; the paper studies their interactions and derives geometric formulae, extending results of Ara--Maltsiniotis, Verity, and others.

Significance. If the equivalence holds, the work supplies a geometric characterization of (∞,∞)-categories via sheaves on oriented polytopes, together with a proof of the oriented Street--Roberts conjecture. It derives standard higher-categorical operations directly from polytope geometry and generalizes prior results on these operations. The manuscript therefore provides both a new perspective and explicit geometric formulae.

minor comments (3)
  1. The abstract states the main theorem but does not name the precise families of oriented polytopes employed; a short list or reference to the relevant section would improve readability.
  2. Notation for the sheaf condition and the precise categorical axioms recovered should be cross-referenced to the definitions of oriented polytope introduced in the paper.
  3. The generalization of Campion's result would benefit from an explicit statement of which theorem or corollary is being extended.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the geometric approach to (∞,∞)-categories, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a proof that (∞,∞)-categories arise as sheaves on families of oriented polytopes (Street simplices, Gray cubes, etc.), generalizing Campion's work. The abstract and description contain no self-citations to load-bearing prior results by the same authors, no fitted parameters renamed as predictions, and no self-definitional steps where the output is constructed to match the input by definition. The central claim is a mathematical equivalence established via geometric constructions and sheaf conditions, with no reduction to the paper's own inputs exhibited in the provided text. This is the expected outcome for a self-contained proof paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on a newly introduced definition of oriented polytope and on the assertion that certain families of these polytopes yield exactly the (∞,∞)-categories via the sheaf condition; no free parameters are mentioned.

axioms (1)
  • standard math Standard axioms and definitions of higher category theory and sheaf theory on simplicial or cubical sets
    The paper builds directly on existing frameworks in algebraic topology and category theory.
invented entities (1)
  • oriented polytope no independent evidence
    purpose: To provide a common geometric object that includes both oriented simplices and oriented cubes
    Newly formulated in the paper to state the oriented conjecture.

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Reference graph

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