15 problems

Let $\Catn$ denote the category of $(\infty,n)$-categories, let $\Shv(\grid)\univ$ denote the category of univalent sheaves on the grid, and let … be the cubical nerve. For $n=2$,…

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…

Let $\mathscr{C}$ be an $(\infty,2)$-category, and let $\mathscr{C}_v$ and $\mathscr{C}_h$ denote its vertical and horizontal inclusions as double $\infty$-categories. There is a s…

Let $\mathcal{C}$ be a category, and let $S(n)$ and $I(n+1)$ denote the objects occurring in the maps $s^n:*\rightarrow S(n)\otimes I(n+1)$. Let…

Let ${\mathcal A}$, ${\mathcal B}$, and ${\mathcal C}$ be double categories, let $Dbl_{hop}(-,-)$ denote the relevant category of double functors, let $Lax_{hop}(-,-)$ denote the r…

Let $B$ be a category, let $B^*$ denote its relevant indexing category, and let $\Phi,\Psi:B^*\to\operatorname{\mathbf{Cat}}$ be $\pi_2$-indexings. Write \textbf{pi2…

Tricategorical univalence conjecture. The tricategory of univalent weak double categories is univalent.

Enriched profunctor conjecture. (Enriched) categories, (enriched) functors and (enriched) profunctors assemble into a univalent weak double category, but not into a univalent doubl…

Embedding conjecture. There is a fully faithful embedding of weak double categories into Verity's double bicategories with essential image given by double bicategories in which the…

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

Conjecture. These will be closely related to regular fibrations and subobject fibrations.

A black-boxing construction assigns to each open Markov process a linear relation between its boundary vector spaces, and the resulting symmetric monoidal double functor is … Here…

Let $\mathbb{D}$ be a double category in $\mathbf{FbrCat}^{\mathcal{Q}}$ that is Cartesian, admits Kleisli objects for monads, and has a discrete Kleisli object for every monad. He…

Let $\mathbb{D}$ be a Cartesian and fibrant double category. An object of $\mathbb{D}$ is called discrete when the two pullbacks defining discreteness satisfy the Beck–Chevalley co…

Let $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$ be the double category of open Petri nets, and let $\mathbb{O}\mathbf{neWay}(\mathsf{Petri})$ be the full sub-double category whose hor…