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1. Classification of $$\mathbb{Z}/2\mathbb{Z}$$-quadratic unitary fusion categories (with an appendix by Ryan Johnson, Siu-Hung Ng, David Penneys, Jolie Roat, Matthew Titsworth, and Henry Tucker)

   https://doi.org/10.4171/QT/201  

   Edie-Michell, Cain; Izumi, Masaki; Penneys, David (February 2024, Quantum Topology)

   A unitary fusion category is called $$\mathbb{Z}/2\mathbb{Z}$$-quadratic if it has a $$\mathbb{Z}/2\mathbb{Z}$$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $$\mathbb{Z}/2\mathbb{Z}$$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostrik's results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larson's rank-finiteness bound for $$\mathbb{Z}/3\mathbb{Z}$$-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information. 

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2. The Simplicial Coalgebra of Chains Under Three Different Notions of Weak Equivalence

   https://doi.org/10.1093/imrn/rnae031  

   Raptis, George; Rivera, Manuel (July 2024, International Mathematics Research Notices)

   Abstract We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring $$R$$. The weak equivalences are given by: (1) an $$R$$-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an $$R$$-homology equivalence between universal covers, and (3) $$R$$-homology equivalences. Analogously, for any field $${\mathbb{F}}$$, we construct three model structures on the category of connected simplicial cocommutative $${\mathbb{F}}$$-coalgebras. The weak equivalences in this context are (1′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3′), we prove that, when $${\mathbb{F}}$$ is algebraically closed, the simplicial $${\mathbb{F}}$$-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when $${\mathbb{F}}$$ is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of $${\mathbb{F}}$$. 

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3. Stable Isomorphisms of Operator Algebras

   https://doi.org/10.1093/imrn/rnad146  

   Kakariadis, Evgenios_T A; Katsoulis, Elias G; Li, Xin (July 2023, International Mathematics Research Notices)

   Abstract Let $${{\mathcal{A}}}$$ and $${{\mathcal{B}}}$$ be operator algebras with $$c_{0}$$-isomorphic diagonals and let $${{\mathcal{K}}}$$ denote the compact operators. We show that if $${{\mathcal{A}}}\otimes{{\mathcal{K}}}$$ and $${{\mathcal{B}}}\otimes{{\mathcal{K}}}$$ are isometrically isomorphic, then $${{\mathcal{A}}}$$ and $${{\mathcal{B}}}$$ are isometrically isomorphic. If the algebras $${{\mathcal{A}}}$$ and $${{\mathcal{B}}}$$ satisfy an extra analyticity condition a similar result holds with $${{\mathcal{K}}}$$ being replaced by any operator algebra containing the compact operators. For nonselfadjoint graph algebras this implies that the graph is a complete invariant for various types of isomorphisms, including stable isomorphisms, thus strengthening a recent result of Dor-On, Eilers, and Geffen. Similar results are proven for algebras whose diagonals satisfy cancellation and have $$K_{0}$$-groups isomorphic to $${{\mathbb{Z}}}$$. This has implications in the study of stable isomorphisms between various semicrossed products. 

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4. On dualizability of braided tensor categories

   https://doi.org/10.1112/S0010437X20007630  

   Brochier, Adrien; Jordan, David; Snyder, Noah (March 2021, Compositio Mathematica)

   null (Ed.)

   We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $$q$$ . 

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5. Rank-finiteness for modular categories

   https://doi.org/10.1090/jams/842  

   Bruillard, Paul; Ng, Siu-Hung; Rowell, Eric; Wang, Zhenghan (July 2016, Journal of the American Mathematical Society)

   null (Ed.)

   We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category C \mathcal {C} with N = ord ( T ) N= \textrm {ord}(T) , the order of the modular T T -matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension D 2 D^2 in the Dedekind domain Z [ e 2 π i N ] \mathbb {Z}[e^{\frac {2\pi i}{N}}] is identical to that of N N . 

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