116 problems

Let $\mathsf{R}$ be a retrodiction, or quantum Bayesian inversion, assignment for normal quantum processes between von Neumann algebras equipped with faithful states, satisfying th…

For every finite-dimensional semisimple and cosemisimple Hopf algebra $H$ over a field $k$, the exponent of $H$ divides its dimension: $\exp(H)\mid\dim_k H$.

Let $\mathrm{LagRel}_k^{\circ}$ be the prop generated by the eight morphisms … The theorem preceding the conjecture gives two extraspecial commutative Frobenius monoids and two bim…

Let ${\cal H}$ be the Heegaard category, and let $T(V)$ denote the relevant target associated with a vector space $V$. Residual representability conjecture. If $a$ and $b$ are dist…

Giraud's theorem for Segal topoi. $A$ is a $\mathbb{U}$-Segal topos if and only if:

Let a directed $A_{\infty}$-pre-category be an $A_{\infty}$-pre-category whose objects are indexed by ${{\bf Z}}$ with transversal sequences ordered by the indices, and consider…

Let an $A_{\infty}$-pre-category be understood with the equivalence relation defined in the source, and let an $A_{\infty}$-category with strict identity morphisms be a non-unital…

Let $\2C$ be a braided tensor $C^*$-category of finite dimension, and let $\2Z(\2C)$ denote its center. A modular extension of $\2C$ is a modular category $\2M$ containing $\2C$ as…

Let $\Gamma$ and $\Gamma'$ be lattices with quadratic forms $q$ and $q'$, let $g:(\Gamma\oplus\Gamma^*,q)\to(\Gamma'\oplus{\Gamma'}^*,q')$ be a lattice isomorphism, and let…

Let $F:A\rightarrow B$ be a morphism of $n+1$-categories, and let $\varphi:A\rightarrow nCAT'$ be a family of $n$-categories over $A$. A direct image is a universal family…

Let $\psi:A\rightarrow nCAT'$ be a functor and let $\mathcal P$ be a property of morphisms from $\psi$ to objects of $nCAT'$. A universal morphism with property $\mathcal P$ is a m…

Suppose $A$ is an $n+1$-category admitting arbitrary direct and inverse limits. A functor $h:A^o\rightarrow nCAT'$ is called representable when it is induced by an object of $A$, a…

Yoneda conjecture. The functors $\alpha$ and $\beta$ are fully faithful. This is the higher-categorical analogue of the corresponding fact for $n=0$; the paper gives no proof or re…

Let $\mathsf{T}_{\mathsf{INC}}$ denote the tangent endofunctor on $\mathsf{INC}(\mathcal{C})$, and suppose either preceding existence theorem applies, so that the universal Lincs c…

Let $\textup{\textsf{T}}$ be a finite discrete time category, let $(\mathcal{C},\mathcal{G},\Omega)$ be a spined sd-category with the additional structure required for persistent n…

Let $\mathbf{Caus}$ be the category of finite acyclic directed graphs equipped with a chosen topological order, with order-preserving graph homomorphisms or refinements as morphism…

Balmer's Nerves of Steel Conjecture. The map $\phi$ is bijective. This conjecture seeks to reconcile the homological and triangular spectra of a rigid tensor-triangulated category.…

Path-category realization conjecture. For every spectroid $\mathcal{C}$, there exists a choice of $(\mathbb{X},\Gamma{/}{\sim})$ such that $\mathcal{C}$ is the $\Bbbk$-linear path…

Let $R\colon \Cat^L_\infty\hookrightarrow \Cat_\infty$ be the embedding of the category of presentable $\infty$-categories and colimit-preserving functors into the category of…

Let $\mathcal{C}$ be a Gray monoid. A braided and symmetric pseudomonoid additivity conjecture. Every braiding on $\mathcal{C}$ induces an equivalence … of 2-categories, and every…

Cartesianity conjecture. If $C$ is an infinity-category, then

Let $\mathsf {BaireMeas}$ be the category of Baire measurable spaces, and let $\mathsf {BaireStoch}$ be its associated Markov category of kernels. A Markov functor is understood to…

Let $n\mathsf{Pr}$ be the $1$-category of presentable $n$-categories, defined as the filtered colimit … where $n\mathsf{Pr}^{\kappa}$ is the category of $\kappa$-compactly generate…

Cartesian multicategories as retromodels. The category of cartesian models is the category of retromodels of $\ifstrequal{S}{1}{\mathds{1}}{\mathbb{S}}^\mathrm{op}$ in…

Malleability conjecture. The following conditions are equivalent: