36 problems

Let $\widetilde{\mathcal{N}}$ be the Springer resolution, let $u(G_q)$ be the small quantum group, let $\operatorname{PrinBlock}_{\mathrm{dg}}(u(G_q))$ be its principal block, and…

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…

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…

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…

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

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.

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

Associativity conjecture. The $\mathbb{Z}[t^{\pm1/2}]$-module $K(\mathscr{C})_t$ with this operation is a $\mathbb{Z}[t^{\pm1/2}]$-algebra, and hence gives a not necessarily commut…

Trace-unit conjecture. The category $[\mathcal{S}^{\operatorname{op}},\mathbf{Vec}_{\Bbbk}]$ is symmetric monoidal, and its unit is equivalent to the trace of the underlying semigr…

Let $\mathsf{C}$ be a Markov category with supports, and let $p$ and $q$ be morphisms in $\mathsf{C}$. Write $S_f$ for the support object of a morphism $f$. Support tensor-product…

Let $\mathbb{A}$ be a distributive monoidal category. Write \mathbf{V}\,{}^\ulcorner_\llcorner\!{\mathbb{A}}\!_\lrcorner^\urcorner^\oplus and…

Let $𝕄$ be a monoidal category, and let $𝕄^{\mathrm{Rev}}$ denote its reverse monoidal category, with tensor product … The objects, 1-cells, 2-cells and 3-cells of the tricategory…

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 ${\mathscr{C}}$ be a monoidal category fulfilling a few mild conditions. Let $Z$ be a braided commutative monoid in the left weak monoidal centre of ${\mathscr{C}}$, and let…

Let $G$ be a finite group, let $\tau_h$ denote the transgression map, and let $\omega\in Z^{n+2}(G,\mathbf{k}^{\times})$. Suppose that a suitable notion of an $n$-category has been…

Let $\mathsf{FM}_2^{\mathrm{Ext}}$ be the Ext-enhanced category of free-monodromic complexes, and let $\mathrm{Tilt}_2^{\mathrm{Ext}}$ be the full subcategory consisting of…

Optic biadjunction conjecture. The functor

Let $\mathbf{CPol}$ be the category of constructible polygraphs, with monoidal structures given by the lax Gray product $\scaleobj{0.75}{\boxtimes}$ and the join $\star$, and let…

Let $\Xi$ be the universe of categories enriched in pointed DCPOs and Scott-continuous functors between them, respecting finite coproducts and initial objects. Let $\Theta$ be the…

Let $\mathcal{C}$ be a rigid monoidal abelian category with a system of renormalized $r$-matrices, and let $(\{\mathcal{F}_i\}_{i \in I}, \widetilde{B})$ be a monoidal seed, where…

Finite-dimensional Frobenius algebra conjecture. Then $A$ admits a unit for $\mu$ and a counit for $\delta$, and is, in fact, a Frobenius algebra.

Let $\mathcal{X}$ and $\mathcal{C}$ be left closed monoidal categories, meaning that tensoring on the left has a right adjoint. Let there be a strong action of $\mathcal{X}$ on…