12 problems

Branching-homology vanishing conjecture. For every integer $n>0$,

Let $I^n$ be the $n$-cube, let $2_n$ be the free $\omega$-category generated by an $n$-morphism, and let $G_n$ for $n>0$ be the oriented $n$-globe, namely the free $\omega$-categor…

Let $\mathcal{C}$ be an $\omega$-category and let $I^1$ be the interval in the biclosed monoidal structure on $\omega\operatorname{Cat}$. For either choice of sign, consider the…

Let $\mathcal{C}$ and $\mathcal{D}$ be free $\omega$-categories, let $f,g:\mathcal{C}\to\mathcal{D}$ be homotopic non $1$-contracting $\omega$-functors, and let…

Let $\mathcal{C}$ be an $\omega$-category, let $I^n$ denote the $n$-dimensional cube, let $A$ and $B$ be composable $n$-morphisms, and let $\widehat{\mathcal{C}}$ denote the modifi…

Model structure conjecture. A similar model structure should exist for weak $\omega$-groupoids and weak $\omega$-categories.

Consider weak $\omega$-categories equipped with weak equivalences, and let the groupoidal core denote the substructure consisting of the invertible higher-categorical data. A weak…

Let a freely generated monoidal weak $\omega$-category be generated by a finite computad, and let $e$ denote its monoidal unit. Its iterated monoidal products of $e$ are also part…

A type-theoretic $\omega$-category is a model of the type theory defined in the paper, while a Grothendieck–Maltsiniotis $\omega$-category is a functor…

Let $\operatorname{cd}(t)$ denote the coherence depth of a term $t$, namely the number of nested coherences, with $\operatorname{cd}(x)=0$ and…

Let $\mathcal{S}_{\mathrm{cat}}$ be the category of type-theoretic models described in the paper, and let $\mathcal{T}$ be the canonical coherator for Grothendieck–Maltsiniotis…

Let $B^{\bullet}_{C}$ be the coglobular complex of colored $\omega$-operads, with $B^{0}_{C}$ the Batanin $\omega$-operad for weak $\omega$-categories, and let…