18 problems

Antieau–Gepner–Heller's theorem of the heart conjecture. The map

Let $\mathcal{A}$ be a smooth $\mathbb{C}$-linear stable $\infty$-category. The finite Quillen-Lichtenbaum dimension conjecture. Then … Here $\dim_{\mathrm{QL}}\mathcal{A}$ is the…

Let $\mathbb{X}$ be an $\mathbb{A}_n$-schober, and let $\mathbb{X}_{11 \cdots 1}$ denote its underlying stable $\infty$-category. Coherent braid-action conjecture. $\mathbb{X}$ def…

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 $\sC$ be a small stable $\infty$-category equipped with a bounded t-structure. A spectrum is connective when its negative homotopy groups vanish. Antieau–Gepner–Heller's connec…

Let $\sC$ be a small stable $\infty$-category equipped with a bounded t-structure, and let $\sC^\heartsuit$ denote its heart. Write $\sD^b(\sC^\heartsuit)$ for the bounded derived…

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

Let $\operatorname{Sph}$ be the $\infty$-category formed by spherical adjunctions, and let ${\mathcal{S}}t_{\Lambda_\infty}^{\operatorname{adm},\,2\text{-Segal}}$ be the $\infty$-c…

Let $({\operatorname{Cat}}_\infty)_{\mathbb{D}\Delta}$ and $({\operatorname{Cat}}_\infty)_{\mathbb{L}\Lambda_\infty}$ denote the $\infty$-categories of $2$-simplicial and $2$-parac…

Let $F:{\boldsymbol{\text{A}}}\to{\boldsymbol{\text{B}}}$ be an exact functor between stable $\infty$-categories, and let $S_\bullet(F)$ denote its relative $S$-construction. The c…

Cotwist conjecture. There is a commutative diagram in $\operatorname{LinCat}_R$ identifying $T_{\operatorname{Fun}(S^n,\operatorname{RMod}_R)}$ under the equivalence…

Sphericality conjecture. For any good stratification $A$, this pullback-limit adjunction is spherical and arises as the restriction of a spherical adjunction involving the sheaf in…

Height Finiteness Conjecture. For every $\mathcal{C}\in\Pr_{\mathrm{st}}^{\mathrm{\infty\text{-}semiadditive}}$, the full subcategory $\mathcal{C}_{\infty}$ is trivial.

Bootstrap Conjecture. Every stable $p$-local presentable $1$-semiadditive $\infty$-category is in fact $\infty$-semiadditive.

A presentable stable $\infty$-category is an $\infty$-category that is presentable and stable, and $1$-semiadditive and $\infty$-semiadditive refer to the corresponding higher semi…

Heart-equivalence conjecture. This natural map should be an equivalence of nonconnective $K$-theory spectra. The assertion would identify the $K$-theory of a boundedly $t$-structur…

Let $E$ be a small stable $\infty$-category equipped with a bounded $t$-structure. Write $\mathrm{K}_{-n}(E)$ for its negative nonconnective $K$-groups. Bounded t-structure vanishi…

Nadler–Tanaka's point-generation conjecture. The object $pt\in\Lag_{pt}(pt)$ is a generator, every object of $\Lag_{pt}(pt)$ is a finite colimit of objects $pt[k]$ for…