65 problems

Contractibility conjecture. The space

Bondal–Polishchuk conjecture. This action should always be transitive, up to independent shifts of the $n$ exceptional objects.

Freyd's Generating Hypothesis for ${\mathcal{F}}_p$. The functor $\pi_*$ is faithful. Therefore, ${\mathcal{M}}_p$ is a Freyd category, with ${\mathcal{F}}_p$ as its full subcatego…

Let $\mathbb D$ be a triangulated derivator, let $\mathbb E$ be a Verdier quotient of $\mathbb D$, and let $K$ denote derivator K-theory. A Verdier quotient is a triangulated local…

Let $\mathcal W$ be a saturated Waldhausen category satisfying the cylinder axiom, and let $\mathbb D_\mathcal W$ be the associated left pointed derivator. The comparison map … is…

The telescope conjecture. Every smashing localizing subcategory $\mathcal{L}$ is generated by compact objects of $\mathcal{T}$.

Margolis Uniqueness Conjecture. If there is an exact equivalence

Coefficient comparison conjecture. There are equivalences of triangulated categories

Let $\mathcal{T}$ be a compactly generated triangulated category, and let $\mathcal{S}$ be a smashing subcategory, meaning a subcategory whose inclusion admits a coproduct-preservi…

Let $\mathsf{T}$ be a compactly generated triangulated category. A localization of $\mathsf{T}$ is smashing if its kernel is a strictly localizing subcategory and its corresponding…

Telescope Conjecture. Then

A triangulated category $\mathsf{D}$ is Hom bounded if for every pair of objects $A,B\in\mathsf{D}$, the dimensions of $\operatorname{Hom}(A,B)$ are bounded independently of the ch…

Extended Uniqueness Conjecture. If there is an exact equivalence

Enhancement-free reconstruction conjecture. It has long been conjectured that, at least in some cases, $\cb$ can be reconstructed from $\ca$ without using an enhancement.

Let $X_1$ and $X_2$ be smooth projective schemes over a field $\Bbbk$. An exact $\Bbbk$-linear functor … is Fourier–Mukai if there exists an object…

Generalized telescope conjecture. The assertion is exactly the stable telescope conjecture, or equivalently its ideal-theoretic reformulation, with $\mathbf{t}$ a t-structure in…

Krause's reformulated telescope conjecture. The ideal $I_{\mathbf{t}}$ is generated by identity morphisms of objects in $\mathcal{D}^c$.

Triangulated homotopy category conjecture. The category $\operatorname{Ho}(\mathcal T)$ is always a triangulated category for an arbitrary proper class $\xi$ of triangles.

Let $\mathsf{T}$ be a tensor-triangulated category with compact objects $\mathsf{T}^{\mathrm{c}}$. Write $\mathsf{Spc}^{\mathsf{h}}(\mathsf{T}^{\mathrm{c}})$ for the space of homol…

Faithfulness conjecture. The action of the Artin–Tits group $\mathbb B$ on the triangulated category $\mathcal D$ is faithful.

Let $\mathcal T$ be a triangulated category and let $\mathcal H:{\mathrm{Alg}_{\ell}}\to\mathcal T$ be an excisive, homotopy invariant, matricially-stable and infinitely additive f…

Ito's conjecture. The Fourier–Mukai spectrum of $\mathcal T$ should coincide with its Serre spectrum:

Let $X$ be a smooth projective rational surface, and let $D^b\bigl(\operatorname{Coh}(X)\bigr)$ denote its bounded derived category of coherent sheaves. The category has finite len…

Let $\mathcal{D}$ be a triangulated category, and let $\operatorname{Stab}(\mathcal{D})$ denote its stability manifold with respect to a fixed lattice. Folklore conjecture. If … th…

Let $\mathcal T$ be a triangulated category with $X\in\operatorname{FM}\mathcal T$. A topological space is locally spectral if it has an open cover by spectral spaces, and locally…