82 problems

Let $\mathcal{C}$ be a braided generalized near-group fusion category. Write $R(m,n)$ for the corresponding generalized near-group fusion ring, with $m,n\in\mathbb{Z}_{\geq 1}$, an…

Rigidity conjecture. (i) Any fusion ring has at most finitely many realizations over ${\bf k}$, up to equivalence (possibly none). (ii) The number of tensor functors between two fi…

Let $l\geq 7$ be an integer not divisible by $3$, and let $q$ be the parameter defining the braided finite tensor category $\mathcal{C}(G_2,G_2(a_1),l,q)$. Write $T(\omega_1)$ and…

Let $N$ and $k$ range over positive integers, and consider all non-pointed étale algebra objects in the categories $\mathcal{C}(\mathfrak{sl}_N,k)$. Their super-transitivity is the…

Let $N$ and $k$ range over positive integers, and let $\mathcal{C}(\mathfrak{sl}_N,k)$ denote the corresponding categories. Let $\mathcal{A}$ be the set of all non-pointed étale al…

Let $\mathcal{C}$ be a modular, ribbon or braided fusion category over $\mathbb{C}$. A category is Hermitian over a field $K$ when it admits the corresponding Hermitian structure d…

Let $\mathds{k}$ be a field, and let $\widehat{\mathbf{SMF1C}}^{\mathrm{dom,faith}}_{\mathds{k}}$ be the $(2,1)$-category of extended symmetric (multi)fusion $1$-categories over…

Let $G$ be a group. For a multifusion category $\mathcal{C}$ with a $G$-action, write $\mathcal{C}^G$ for its equivariantization and $\mathbf{Mod}(\mathcal{C})$ for its delooping.…

2-Tillmann conjecture. An object of the symmetric monoidal $3$-category $\mathbf{2Kar}$ is $1$-dualizable if and only if it is a finite semisimple $2$-category.

Let $G$ be a symmetry group, let $\mathcal{S}_G$ be a $G$-graded super fusion category with trivial $\omega_2$, and let $\mathcal{S}_1$ be its trivial grading sector. The category…

Naidu–Rowell Property $F$ conjecture. The braid group representations associated with any object in a weakly integral braided fusion category have finite images.

Let $X$ be a sufficiently nice parameter space, such as a CW complex, let $\mathcal{C}$ be a fusion category, and let $\mathcal{M}$ be a fixed $\mathcal{C}$-module category with co…

Let $\mathcal{C}$ be a fusion category, let $\mathcal{M}$ be a $\mathcal{C}$-module category, and denote the corresponding gapped phase by $\mathcal{T}_{\mathcal{M}}^{\mathcal{C}}$…

Let $A$ be a finite abelian group. Write $\mathcal{W}itt(A,\mathsf{triv})$ for the Witt group associated to $A$ with trivial syllepsis, let $\mathcal{W}itt$ denote the ordinary Wit…

Wall-fusion associativity conjecture. Fusion through $\CC$ satisfies

Condensable-algebra fusion conjecture. Their relative tensor product satisfies

Separable-algebra decomposition conjecture. There is a separable algebra $B'\in\CM$ such that

Homological modularity conjecture. The category $\mathsf{RFC}(\mathfrak{sl}(N),M_3)$ is modular if and only if

Let $\cal{C}$ be a modular tensor category of rank $k$ with modular data $S$ and $T$. A modular invariant is a matrix $M$ satisfying $MS=SM$, $MT=TM$, $M_{ab}\in\mathbb{N}$ for all…

Let $\mathcal C$ be a modular tensor category, and let $\operatorname{FPdim}(\mathcal C)$ denote its Frobenius–Perron dimension. Solvability conjecture. If…

Let $\mathcal{B}$ be a bicommutant category admitting an absorbing object whose endomorphism algebra is a hyperfinite $\mathrm{II}_\infty$ or $\mathrm{III}_1$ factor. Assume that ……

Let $C_p$ be the cyclic group of prime order $p$, let $R(C_p,\ell)$ be the near-group fusion ring with group of invertible elements $C_p$ and parameter $\ell$, and call it categori…

Integration conjecture. For every such $\Sigma$ and $A$, there is an equivalence

Let generalized Asaeda–Haagerup categories be de-equivariantizations of generalized Haagerup categories for the groups $\mathbb{Z}_{4m}\times\mathbb{Z}_2$ with $\epsilon_{(0,1)}$ t…

Let $\mathcal{C}$ be a fusion category, and let an associative zesting of $\mathcal{C}$ induce a zesting on its center $\mathcal{Z}(\mathcal{C})$. Induced zesting conjecture. The i…