Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.10636

Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology

Diogo Andrade·Julia Bierent·Jennifer Brown·Tudor Caba·Matthew Cellot·Jonathan Davies·Adrien DeLazzer Meunier·Jannik Gröne·Benjamin Haïoun·Alea Hofstetter·Theo Johnson-Freyd·David Jordan·Tessa Kammermeier·Patrick Kinnear·Cameron Krulewski·Theodoros Lagiotis·Leon Liu·Adrià Marín Salvador·Nivedita·David Reutter·Lorenzo Riva·Iordanis Romaidis·Jack Romo·Sean Sanford·Michail Tagaris·Daniel Teixeira·Jackson van Dyke·Matthias Vancraeynest·Chetan Vuppulury·Matthew Yu·Markus Zetto
View original source ↗Read source document ↗Export metadata ↓

Abstract

These lecture notes are the product of a week-long learning workshop on the work of Johnson-Freyd and Reutter on the problem of the existence of minimal nondegenerate extensions of braided fusion categories (arXiv:2105.15167). They recount the mathematical arguments of the original paper from an expository angle, with background material covering the algebra and homotopy theory required to understand the statement and follow the proof. The notes are aimed at newcomers to the field of (braided) fusion 1- and 2-categories.

Explore related subjects

math.QAmath.CTmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Diogo Andrade, Julia Bierent, Jennifer Brown, Tudor Caba, Matthew Cellot, Jonathan Davies, Adrien DeLazzer Meunier, Jannik Gröne, Benjamin Haïoun, Alea Hofstetter, Theo Johnson-Freyd, David Jordan, Tessa Kammermeier, Patrick Kinnear, Cameron Krulewski, Theodoros Lagiotis, Leon Liu, Adrià Marín Salvador, Nivedita, David Reutter, Lorenzo Riva, Iordanis Romaidis, Jack Romo, Sean Sanford, Michail Tagaris, Daniel Teixeira, Jackson van Dyke, Matthias Vancraeynest, Chetan Vuppulury, Matthew Yu, Markus Zetto. 2025-09-12. Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology. https://arxiv.org/abs/2509.10636

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Brunnian braids and the inclusion from double shuffle Lie algebra to Kashiwara-Vergne Lie algebra

Schneps \cite{Schneps2012,Schneps2025} and Enriquez-Furusho \cite{EF4} proved that the double shuffle Lie algebra $\mathfrak{dmr}_0$ embeds into the Kashiwara--Vergne Lie algebra $\mathfrak{krv}_2$. We give a Brunnian braid interpretation of a related embedding into the symmetric Kashiwara--Vergne Lie algebra $\mathfrak{krv}_2^{\mathrm{sym}}$. More precisely, the map \[ φ\longmapsto \bigl(φ(-x_0-x_1,x_0),φ(-x_0-x_1,x_1)\bigr) \] defines an injective Lie algebra homomorphism from the subalgebra of $\mathfrak{dmr}_0$ satisfying the condition \[ [x_0,φ(-x_0-x_1,x_0)] +[x_1,φ(-x_0-x_1,x_1)]=0 \] into $\mathfrak{krv}_2^{\mathrm{sym}}$. The proof reformulate the double shuffle and symmetric Kashiwara--Vergne relations through abelianizations of Brunnian Lie algebras associated with the disk and punctured disks. We generalize this inclusion in two directions. First, replacing these abelianizations by higher lower central series quotients yields generalizations of relations and implications among them. Second, we establish explicit identities relating the linear pentagon defect to the stuffle coproduct, the divergence map, and the necklace cobracket.

math.QA↗
arXiv2026

A $q$-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension

Let $H$ be a pointed Hopf algebra of square-free dimension over an algebraically closed field of characteristic $p>0$. We prove that either $H$ is a group algebra or $\dim H/|\G(H)|=p$, and that in the latter case $H$ belongs to exactly one of two explicit families of rank-one pointed Hopf algebras. We develop a truncated $q$-Weyl freeness principle for finite-dimensional Nichols algebras of quandle type. If $V=\bigoplus_{x\in X}\K e_x$, $X'\subsetneq X$ is a nonempty subquandle, $V'=\bigoplus_{x\in X'}\K e_x$, and $s\in X\setminus X'$, then $\mathcal B(V)\simeq\K[e_s]/(e_s^{m_s})\otimes C_{s,X'}\otimes\mathcal B(V')$ for some graded vector space $C_{s,X'}$, where $m_s$ is the nilpotency order of $e_s$; in particular, $(m_s)_z\,\mathcal H_{\mathcal B(V')}(z)\mid\mathcal H_{\mathcal B(V)}(z)$. In the non-group case, this yields a $p^2$-divisibility obstruction that rules out noncentral support for the infinitesimal braiding. Together with a graded-dual argument, the resulting rank-one reduction forces the diagram of $H$ to have dimension $p$.

math.QA↗