91 problems

Pivotality conjecture. Every fusion category is pivotal.

Let $V_1,V_2,V_3$ be objects of the shifted category $\mathcal{O}_{sh}$. Associativity conjecture. There exists an isomorphism … Thus, $k$-fold tensor products in…

Decategorification conjecture. The decategorification of ${\mathcal{B}}_W$ is equal to $B_W$.

Let $M$ be a symmetric monoidal closed model category satisfying suitable hypotheses, such as being cofibrantly generated and left proper. Let $E_{\infty}\text{-}Alg(M)$ be the cat…

Let ${\mathcal C}$ be a rigid monoidal category with finitely many irreducible objects. A module category over ${\mathcal C}$ is called indecomposable if it is not equivalent to a…

Let $R$ be a finite-dimensional unital based ring, and let ${\mathcal C}$ be a rigid monoidal category whose Grothendieck ring is isomorphic to $R$. Ocneanu rigidity. For a fixed…

Weak-equivalence conjecture. The semi-strict framework with strict interchange should be equivalent to a fully weak version.

Refined embedding conjecture for quantum channels. The Kleisli category of the induced comonad on $\operatorname{Fun}(\mathcal{S},\mathbf{H}_{\mathrm{sm}})$, with Day convolution,…

Let $\mathcal{B}$ be a rigid monoidal category regarded as an object of $\operatorname{Alg}_1(\Pr)$. An $SO(2)$-homotopy-fixed-point on $\mathcal{B}$ consists of an isomorphism of…

Let $\mathfrak C$ be a monoidal $\mathbb K$-linear 2-category, subject to suitable assumptions. Write $\mathfrak C^\times$ for its subgroupoid of invertible objects, and let…

Ferrer’s conjecture. Any such $\mathcal{C}$ admits an $\operatorname{O(n)}$-volutive structure that restricts on $\mathcal{C}^\times$ to the canonical $\operatorname{O(n)}$-action…

A module over $\mathbf{Gr}(\mathscr{F})$ is called realizable if it is isomorphic to $[\mathcal{M}]_\oplus$ for some transitive subcategory of $\mathrm{add}(\mathscr{F}\cdot L)$, w…

Let $\mathfrak{g}$ be the Lie-theoretic datum of the paper, let $\mathcal O_{\mathrm{sh}}$ be the shifted category O, and let $R=\bigoplus_{\nu\in Q_+}R_\nu$ be the associated KLR…

Suppose that $\mathtt{C}$ and $\mathtt{D}$ are coboundary monoidal categories, for example $\mathtt{C}=\mathfrak{g}\text{-}\mathtt{Crys}$ and…

Weimar–Woods conjecture. Two graded contractions $\varepsilon$ and $\varepsilon'$ are isomorphic if and only if they are equivalent via normalization:

Let $n$ be a positive integer, and let a symmetric monoidal $n$-category with duals on all levels mean a symmetric monoidal $n$-category equipped with duality data at every level.…

Let $\mathcal{B}$ be a monoidal bicategory. For each monoid object in $\mathcal{B}$, consider lax monoidal $1$-cells and monoidal $2$-cells. Iterated lax monoid conjecture. It shou…

Coherence conjecture. All such maps defined as composites of associators, unitors, and interchange morphisms are equal.

Let $cat$ be the monoidal category and let $mu([x,y])$ denote the object associated with $[x,y]$ by the mutation appearing in Theorem. An affinization conjecture. The object…

Wiring-diagram multicategory conjecture. The multicategory of wiring diagrams is precisely the produoidal category of lenses once dependencies are no longer tracked explicitly.

Deterministic sub-distributive plasma conjecture. If $R$ is a deterministic plasma with a sub-distributive monoidal structure, then $\hat{H}R$ is canonically an $\mathbb{F}_1$-alge…

Automorphism-detection conjecture. If $M$ is not free, then $M$ cannot be fixed by all algebra automorphisms of $A$.

The converse Morita conjecture. If

Let $\mathscr{C}_{\mathbb{Z}}$ be the relevant representation category, let $K(\mathscr{C}_{\mathbb{Z}})_t$ be its $t$-deformed Grothendieck module, and let…

Let $J$ be the indexing set for a PBW-theory, let $d\in\mathbb{N}$, and let $\boldsymbol{\epsilon}=(\epsilon_1,\ldots,\epsilon_d)\in J^d$. Write $M(\boldsymbol{\epsilon})$ for the…