20 problems

A cocomplete abelian category is an abelian category admitting all small colimits, and a covering class is a class of objects for which every object has a cover. Enochs's conjectur…

Let $\mathcal A$ be an abelian category. Its negative $K$-theory is defined using an enhancement, with $K$-theory understood as Waldhausen $K$-theory of that enhancement. Schlichti…

Let $\mathcal{A}$ be an abelian category with enough injectives, and let $A\in\mathcal{A}$. Consider the contravariant functor $\operatorname{Ext}_{\mathcal{A}}^{1}(-,A)$. Auslande…

Red-path realization conjecture. Every maximal green sequence in $\mathcal{A}$ is induced by a red path in the wall and chamber structure of $\mathcal{A}$.

Extremely strong Artinian conjecture, third version. For every $n\in\mathbb{N}$, the subcategory ${\mathcal{F}}^{\omega-\mathrm{cons}(n)}$ of ${\mathcal{F}}$ is thick.

For a commutative ring $A$, let $A\textrm{-}\mathcal{CAT}^{Mor}$ denote the Morita localization of linear categories and let $A\textrm{-}\mathcal{AB}^{proj}$ be the subcategory of…

Let $\mathcal{A}$ be a Grothendieck abelian category, and let $D(\mathcal{A})$ be its derived category. Let $S$ denote the sphere object, and let…

Given an artinian abelian category $\mathcal A$ and a homomorphism $X \colon K_0(\mathcal A) \to \mathbb R$ positive on the class of every non-zero object, let the iterated HKKP fi…

Let $\mathcal{A}$ be an abelian category with enough projectives, and let $A\in\mathcal{A}$. Consider the covariant functor $\operatorname{Ext}_{\mathcal{A}}^{1}(A,-)$. Dual Auslan…

Let $X$ be a topological space, and let $\mathbf{CS}\left( X,\mathbf{Ab}\right)$ denote the category of classical cosheaves on $X$ with values in abelian groups. The non-abelianity…

Let $(Q,\mathcal{P})$ be a thread quiver and let $\mathcal{C}$ be its path category. Write … for the category of pointwise finite-dimensional representations of $\mathcal{C}$. Here…

Let $X$ be an algebraic variety over $\mathbb{Q}$. An object $M$ of the abelian category $\mathcal{M}_\mathrm{univ}(X,\mathbb{Z})$ is torsion-free if multiplication by every nonzer…

Let $\mathcal{A}$ be an Ext-finite abelian category with Serre duality, and let $(\mathcal{T},\mathcal{F})$ be a torsion pair on $\mathcal{A}$. Denote by $U_{\mathcal{T}}$ the aisl…

Let $\mathcal{M}$ be a small $d$-abelian category. Write $\operatorname{Ind}(\mathcal{M})$ for its ind-completion, and let $\operatorname{Mod}\mathcal{M}$ be the category of functo…

Axiomatization conjecture for extended directories. Then $D$ is an extended directory: there is an object $A$ in a $K_0$-linear abelian category such that

Let $\mathsf B$ be a locally presentable abelian category with a projective generator $P$. Let $\mathsf B_{\operatorname{proj}}$ denote the class of projective objects in…

Hereditary abelian subcategory conjecture. The category $\mathcal C'$ is a $\operatorname{Hom}$-finite hereditary abelian category.

Eilenberg–Watts conjecture. The Eilenberg–Watts theorem should hold in any abelian monoidal category.

Global-dimension extension. This is true for all Abelian categories with coproducts.

Let $\mathcal C$ be a skeletally small abelian category, and let … be the category of exact functors $\mathcal C^{\mathrm{op}}\to\operatorname{Ab}$. Equip $\mathcal A$ with its can…