Related papers: Compact Semisimple Tensor 2-Categories are Morita …
We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension, and hence that every pseudo-unitary super modular tensor category admits a minimal modular extension. This completes the program of…
Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or…
We study simple extensions of pointed finite tensor categories, that is, tensor categories $\mathcal{C}$ admitting an abelian decomposition $\mathcal{C} \cong \mathcal{D} \oplus \mathcal{M}$ where $\mathcal{D}$ is a pointed tensor…
We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG…
We prove that braided fusion categories of Frobenius-Perron $p^mq^nd$ or $p^2q^2r^2$ are weakly group-theoretical, where $p,q,r$ are distinct prime numbers, $d$ is a square-free natural number such that $(pq,d)=1$. As an application, we…
A finite pre-tensor category is a finite abelian category equipped with a right exact tensor product for which every projective object has duals. Finite tensor categories, for which every object has duals, are notable examples. More…
In this note we prove two main results. 1. In a rigid braided finite tensor category over C (not necessarily semisimple), some power of the Casimir element and some even power of the braiding is unipotent. 2. In a (semisimple) modular…
We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep(O(\infty) (formally), Rep(O(N), Rep(Sp(N) or of one of its associated fusion categories. If the braiding is not…
We classify pointed fusion categories C(G, $\omega$) up to Morita equivalence for 1 < |G| < 32. Among them, the cases |G| = 2 3 , 2 4 and 3 3 are emphasized. Although the equivalence classes of such categories are not distinguished by their…
In this paper we give a complete classification of pointed fusion categories over $\mathbb{C}$ of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine…
Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain 2-morphisms in the 2-category of `2-tangles in 4…
We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion…
We study the fusion 3-categorical symmetries for quantum theories in (3+1)d with self-duality defects. Such defects have been realized physically by half-space gauging in theories with 1-form symmetries $A[1]$ for an abelian group $A$, and…
We classify (multi)fusion 2-categories in terms of braided fusion categories and group cohomological data. This classification is homotopy coherent -- we provide an equivalence between the 3-groupoid of (multi)fusion 2-categories up to…
A quasi-schemoid is a small category whose morphisms are colored with appropriate combinatorial data. In this note, Mitchell's embedding theorem for a tame schemoid is established. The result allows us to give a cofibrantly generated model…
A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their…
We consider two families of categories. The first is the family of semisimple quotients of H. Andersen's tilting module categories for quantum groups of Lie type $B$ specialized at odd roots of unity. The second consists of categories…
In the preceeding paper we constructed an infinite exact sequence a la Villamayor-Zelinsky for a symmetric finite tensor category. It consists of cohomology groups evaluated at three types of coefficients which repeat periodically. In the…
Noncommutative near-group fusion categories were completely classified in the previous work of the first named author by using an operator algebraic method (and hence under the assumption of unitarity), and they were shown to be group…
This work is a detailed version of arXiv:0704.0195 [math.QA]. We introduce a new notion of the core of a braided fusion category. It allows to separate the part of a braided fusion category that does not come from finite groups. We also…