27 problems

Let $k$ be an algebraically closed field and let $\left({\mathcal C},\otimes,\mathbf{1}\right)$ be a finite tensor category over $k$. Its cohomology ring is…

Etingof–Ostrik's conjecture. An algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras.

Let $k$ be a field and let $({\mathcal C},\otimes,\mathbf{1})$ be a finite tensor category over $k$. Its cohomology ring is … and each object $X\in{\mathcal C}$ has cohomology…

Let $\mathcal K$ be a finite tensor category, let $\mathsf{C}$ denote its categorical center, and let … be the comparison map. The comparison-map homeomorphism conjecture. For ever…

Let ${\mathcal C}$ be a finite tensor category, let ${\bf 1}$ be its tensor unit, and let $X\in{\mathcal C}$. Write ${\rm Ext}^*({\bf 1},{\bf 1})$ for the graded Ext algebra and…

Let ${\mathcal C}$ be a finite tensor category, let $V\in{\mathcal C}$, let $V^{**}$ denote the double dual of $V$, and let $L_\rho$ be the distinguished invertible object of…

Shimizu's conjecture. Every simple algebra in a finite tensor category is quasi-Frobenius.

Let $\mathbbm{k}$ be an algebraically closed field of characteristic zero, and let $\mathcal{C}$ be a finite tensor category over $\mathbbm{k}$. Write … for the cohomology of an ob…

Central cohomological support conjecture. The central cohomological support satisfies these conditions, and consequently

Let $\mathcal{C}$ be a braided finite tensor category and let $A$ be a commutative algebra in $\mathcal{C}$. Let $\mathcal{C}_A$ be the category of left $A$-modules, and let…

Quasi-Frobenius conjecture. Every simple algebra in $\mathcal{C}$ is quasi-Frobenius.

Finite-type conjecture. Every finitely generated algebra in $mathcal C$ is of finite type.

Exactness conjecture. Every simple algebra in a finite tensor category is exact, without assuming that the algebra is commutative.

Let $\mathbf T$ be a finite tensor category and let $\underline{\mathbf T}$ be its stable category. Let $C^\bullet_{\underline{\mathbf T}}$ be the categorical center of the cohomol…

Let $\mathbf T$ be a finite tensor category, let $\underline{\mathbf T}$ be its stable category, and let … be the surjective continuous map from Theorem B. Vashaw–Yakimov's conject…

Let $\mathbf T$ be a finite tensor category and let $\underline{\mathbf T}$ be its stable category. Denote by $R^\bullet_{\underline{\mathbf T}}$ the cohomology ring and by…

Let $\mathbf T$ be a finite tensor category and let $\underline{\mathbf T}$ be its stable category. Write $C^\bullet_{\underline{\mathbf T}}$ for the categorical center of its coho…

Let $\mathcal{D}$ be a finite tensor category over a field $\mathbf{k}$ of characteristic $2$. Let $o(\mathcal{D})$ and $O(\mathcal{D})$ denote its lower and upper Frobenius orders…

Let $k$ be a field, let $\mathscr{C}$ be a finite tensor category of finite type, and let $\mathscr{M}$ be an exact $\mathscr{C}$-module category. Write…

Let $k$ be a field, let $\mathscr{C}$ be a finite tensor category of finite type, and let $\mathscr{M}$ be an exact $\mathscr{C}$-module category. Write…

Let $k$ be a field and let $\mathscr{C}$ be a finite tensor category over $k$. Say that $\mathscr{C}$ is of finite type when H^{\text{\tinybullet}}(\mathscr{C},\mathbf{1}) is a…

EGNO's conjecture. A pointed finite tensor category is tensor generated by objects of length $2$.

Let $V$ be a simple, non-negatively graded, $C_2$-cofinite vertex operator algebra that is isomorphic to its contragredient module, and let $\operatorname{Rep}(V)$ be the category…

Let $\operatorname{Rep}(\mathcal{V})$ be the representation category of a vertex operator algebra $\mathcal{V}$. Rigidity, pivotality and factorisability conjecture. The category…

Let $\mathcal{C}$ be the braided finite tensor category under consideration, let $\Omega_{\mathcal{C}}$ be the matrix associated with the pairing in Equation, and let…