Symmetry in the cubical Joyal model structure

Doherty, Brandon

doi:10.2140/agt.2026.26.699

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Symmetry in the cubical Joyal model structure

Brandon Doherty

Algebraic & Geometric Topology 26 (2026) 699–734

DOI: 10.2140/agt.2026.26.699

Abstract

We study properties of the cubical Joyal model structures on cubical sets by means of a combinatorial construction which allows for convenient comparisons between categories of cubical sets with and without symmetries. In particular, we prove that the cubical Joyal model structures on categories of cubical sets with connections are cartesian monoidal. Our techniques also allow us to prove that the geometric product of cubical sets (with or without connections) is symmetric up to natural weak equivalence in the cubical Joyal model structure, and to obtain induced model structures for $(\infty, 1)$ -categories on cubical sets with symmetries.

Keywords

cubical sets, $(\infty, 1)$-categories, model categories

Mathematical Subject Classification

Primary: 18N40, 18N60, 18N65

Secondary: 55U35

References

Publication

Received: 27 September 2024

Revised: 23 March 2025

Accepted: 3 April 2025

Published: 11 February 2026

Authors

Brandon Doherty
Department of Mathematics Florida State University Tallahassee, FL United States

This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2026, the article will become Open Access in early 2026. Otherwise, this article (and all 2026 articles) will be available only to paid subscribers.

Volume 26, issue 2 (2026)