82 problems

Naturality and decomposability conjectures. The chain maps $\kappa_X$ and $\eta_X$ are natural, namely, they commute with all 2-morphisms of $\dot{\mathcal{U}}$. Moreover, for…

Let $W$ be the Weyl group, let $S_{\bf d}$ be the subgroup associated with the tensor-product datum ${\bf d}$, and let $H_{\bf d}^i=(B^{\bf d}\otimes C)^{W_i}$ and…

Parabolic Temperley–Lieb relations conjecture. These functors should satisfy the natural isomorphisms corresponding to the defining relations of the Temperley–Lieb category, with…

Let $\cap_{i,n}$ and $\cup_{i,n}$ be the cap and cup functors between the relevant singular categories, and let $R_{i,n}$ be the cone of the adjointness morphism…

For integers $n$ and $k$, let $D^b(\mathcal O_{k,n-k})$ be the bounded derived category of the relevant parabolic category, and define functors … … by…

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\mathbf{\Gamma}^{[a,b]}(\underline{w}_0)$ and…

Let $\underline{w}_0$ be a reduced expression of the longest element and let $[a,b]$ be a finite integer interval. Let $\widehat{\mathbf{\Gamma}}^{[a,b]}(\underline{w}_0)$ be the c…

Positivity conjecture. Every simple module is quantizable. The supplied status evidence says that this conjecture is proved for reachable simple modules in affine types $C_n^{(1)}$…

Let $\mathcal{R}=\mathcal{R}_0$ be an integral fusion ring, and let $(\mathcal{R}_n)$ be the sequence obtained by iterating the integral extension construction of Proposition, star…

Let equation denote the paper's preceding conjectural relation, and let $n$ and $k$ be the rank and generator index appearing there. Higher-rank trace conjecture. Equation also hol…

Let $\beta\in\mathrm{Br}_m$ and let equation denote the paper's stated trace/decategorification equality between type $B$ spin link homology and the spin-colored…

Let $m\geq 2$, let $U'_{-q^2}(\mathfrak{so}_{m})$ be the quantum orthogonal algebra with elements $\mathsf{x}_i^{(k)}$, and let ${\EuScript B}_{m}$ be the relevant monoidal categor…

Let $n\geq 1$ and $m\geq 2$. Write $K_0^\tau(-)$ for the weighted Grothendieck group of a category with involution, let $({\EuScript{B}}_{m}^{n})^\tau$ be the equivariant category,…

Let $n\in\mathbb{N}$, let $(m_i,n_i)\in\mathbb{Z}\times\mathbb{N}$, and let $d_i\geq 1$. For the affine Soergel trace category $\operatorname{Tr}(\mathrm{ASBim}_n)$, write $\star$…

Let $n\in\mathbb{N}$ and let $\operatorname{Tr}(\mathrm{ASBim}_n)^{\operatorname{Kar}}$ be the idempotent completion of the trace category of affine Soergel bimodules. For integers…

Crane–Frenkel conjecture. There exists a categorification of the quantum group $U_q(\mathfrak{sl}_2)$ at roots of unity giving rise to a 4D TQFT.

Let $\Gamma$ be an ADE root system, let $Q$ be an orientation of its Dynkin diagram, and let $\alpha=\sum_{i\in Q_0}n_i\alpha_i$ be a positive root. Set…

Let $K\subset S^3$ be a framed oriented knot and let $X\in\operatorname{Ch}^-(\mathsf{BN}_n)$ be a complex of $n$-strand Bar–Natan 1-morphisms. Let $C(K,V_n)$ denote the $V_n$-colo…

Let $K$ be a framed oriented knot, let $\operatorname{Kar}(\mathcal T\mkern-3mu\mathcal L)^\oplus$ be the additive Karoubi envelope of the Temperley–Lieb category, and let $V_n$ de…

Let $\kappa$ be the coefficient field, let $\operatorname{Ch}^-(\mathsf{BN}_n)$ be the homotopy category of bounded-above complexes of $n$-strand Bar–Natan 1-morphisms, and let…

Let $p$ be a prime. A rigid symmetric monoidal $b\mathbb{Z}d$-linear category is a rigid symmetric monoidal category enriched over $b\mathbb{Z}d$-modules; write $K_0(C)$ for its Gr…

Let $B_{(W,S)}$ be the generalized braid group associated with the Coxeter system $(W,S)$, and let $2$-${\mathcal B}r$ be the monoidal category generated by the Rouquier complexes…

Let $C(S,S):=\operatorname{Hom}_{\mathrm{Ft}(M,\Omega_I)}(S,S)$ and $C(S,F_b)$ be the corresponding Fueter Hom-categories, with the functor $RG_b(\zeta):C(S,S)\to C(S,F_b)$ induced…

qAGH knot invariance conjecture. If $\Bbbk$ is a field of characteristic $0$, then ${\mathit{qAGH}}$ is a knot invariant for any $q$.

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…