52 problems

Limit–terminality conjecture. The following two statements are equivalent:

Comparison conjecture for snucategories. The localization of the category of $n$-snucategories by equivalences is equivalent to the localizations of the categories of weak $n$-cate…

Homotopy hypothesis for snugroupoids. There are functors

Higher-dimensional coherence conjecture. Any small lax $n$-category is laxly $n$-equivalent to a small Gray $n$-category.

Let $\mathcal{C}$ be a non-contracting $\omega$-category freely generated by a precubical set. Let $M_n^{gl^-}(\mathcal{C})$ denote the degree-$n$ elements of the branching semi-gl…

Let $\mathcal{C}$ be a strict non-contracting globular $\omega$-category. Let $C F^{gl}$ be its formal globular complex and $C F^{\pm}$ its formal branching and merging complexes,…

Let $K$ be a precubical set, let $F(K)$ be the free non-contracting $\omega$-category generated by $K$, and let $\widehat{F(K)}$ denote its Kan completion. Let $C^{gl}$ and…

Let $(\bDelta^n)$ denote the oriented simplex, let $\tau_{n-1}(\bDelta^n)$ denote its $(n-1)$-truncation, and let $(\Delta^n)^t$ be the stratified simplicial space obtained from th…

Harpaz's conjecture. The functor $N_{\mathrm{Lurie}}$ induces an equivalence of $\infty$-categories after localizing at weak equivalences.

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…

Let $\mathcal{C}$ be an $(\infty,n)$-category, and let $\mathrm{O}_{n-1}$ and $\Theta_n$ be the indexing shapes used to define the iterated constructions…

Let $\mathfrak{K}_{n}$ and $\mathfrak{M}_{n}$ be the model categories of marked Kan complexes and marked complicial sets, respectively, and let $(i_{\Delta},-_{\Delta})$ be the adj…

Let $n\text{-}\operatorname{Cat}$ denote the category of $n$-categories, let $\operatorname{Surj}_{\omega}$ denote the class of $\omega$-surjective morphisms, and let…

Twisting classification conjecture. There is a canonical isomorphism of anima

Scholze's uniqueness conjecture. The six-functor formalisms produced by the Liu–Zheng construction are exactly those for which every morphism in $P$ is cohomologically proper and e…

Projection-term conjecture. Under these hypotheses, there exists a term $K\vdash p_t:T_t$ with the recursively specified type $T_t$, and its free variables satisfy the displayed un…

Span functoriality conjecture. The $\operatorname{Span}$ construction underlies a limit-preserving $(\infty,2)$-functor from the $(\infty,2)$-category of categories with pullbacks,…

Internal-left-adjoint compact-generation conjecture. If the coevaluation morphism is an internal left adjoint, then $C$ is compactly generated. The paper states that no counterexam…

Let $\mathcal P$ be a “doubly-presentably” symmetric monoidal $(\infty,2)$-category. The $(\infty,1)$-category $\mathcal P^{\mathrm{dbl}}$ of dualizable objects and $\mathcal P$-in…

Let $Z$ be the conical manifold and let $\targetcat$ be the target category appearing in the cone decomposition theorem. For each integer $d\geq 0$, consider the restriction functo…

Sufficiency conjecture. Suppose $\boxed{}$ is a sufficient family of univalent fibrations for $\boxed{}$. Then

Span-preservation conjecture. Let $\boxed{}$ be a univalent discrete cocartesian fibration. Then

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…

Paracyclic–spherical correspondence. There is an equivalence of infinity-categories between the infinity-category of paracyclic stable infinity-categories and the infinity-category…

Nerve completeness conjecture. The $d$-categorical nerve $I$ is a complete and discrete $d$-fold Segal space.