127 problems

Let a weak $n$-category be a higher category whose composition laws are associative and unital only up to coherent invertible higher morphisms. Simpson's weak units conjecture. Any…

An $n$-type is a homotopy type whose homotopy groups vanish in degrees greater than $n$. An $n$-groupoid is an algebraic structure intended to encode homotopical information throug…

Gálvez–Kock–Tonks conjecture. For any decomposition space $X$,

Let a closed term be a term with no free variables, and let a numeral be a canonical term of the type of natural numbers. Voevodsky's homotopy canonicity conjecture. Every closed t…

Let $\mathbb{B}^{0}_C$ be Batanin's globular operad, whose algebras are his models of globular weak $\infty$-categories, and let $\Theta_{\mathbb{B}^{0}_C}$ be its associated theor…

Let $A$ be an $n$-coherent $O$-algebra, and let $x$ be an occupant of an $m$-dimensional $o$-niche. For a frame-competitor $b'$ of the target of $x$, let … be the two virtual…

Homotopy hypothesis. Spaces with vanishing homotopy groups above dimension $n$ are equivalent to weak $n$-groupoids.

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…

Consider the categories of special $\operatorname{cat}^{n-1}\text{-}\operatorname{group}$s and fair $n$-groupoids, together with the strongly contractible faces of a special…

Let $mathcal{K}3$ and $mathcal{H}3$ denote the two semistrictifications of Tamsamani weak $3$-groupoids discussed in the paper, and restrict to the path-connected case. For gener…

A fair $n$-groupoid is Joachim Kock's formalization of a strict $n$-groupoid equipped with a suitable notion of weak identity arrow. Kock-Simpson conjecture. Every homotopy $n$-typ…

A fair $n$-groupoid is Joachim Kock's semistrict structure in which composition is strict while identity arrows are suitably weak. Kock's conjecture. Fair $n$-groupoids should mode…

The discussion concerns a family of $4$-dimensional resource-management cells arising from local computations on proofs, including cells for a weakening followed by a contraction,…

Let $A{\rightarrow} W$ be a map of $n$-groupoids, and let $j$ be a nonnegative integer. For each $k$, write $\pi_k$ for the corresponding homotopy group of an $n$-groupoid. Strong…

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…

Self-dual coherent omega-equivalence conjecture. The augmented chain complex $C$ is the linearisation of a polygraph presenting a self-dual model of the coherent walking $omega$-eq…

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…

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 $n=2$,…

A 3-crossed module is a structure consisting of four groups and six kinds of liftings, as formulated in this paper. Let $\mathcal{C}$ be the category of these 3-crossed modules. Hi…

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

Formal differentiation conjecture. The functor $\operatorname{FmlDiff}$ induces an equivalence between the underlying $\infty$-categories

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…