23 problems

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.

Extremely strong Artinian conjecture, second version. For every $n\in\mathbb{N}$, the functor $\omega_n$ induces an equivalence

Extremely strong Artinian conjecture, first version. For every $n\in\mathbb{N}$, the functor $\omega_n$ induces an equivalence

Noetherianity conjecture. For every $n\in\mathbb{N}$, the functor $P^{\otimes n}$ is noetherian: every ascending chain of subfunctors of $P^{\otimes n}$ stabilizes.

Finite-dimensional values conjecture. Every indecomposable injective functor in $\mathcal{F}$ takes finite-dimensional values.

Omega-adapted nilpotence conjecture. An object of finite type in $\mathcal{F}$ is omega-adapted of height at most $n-1$ if and only if it is $\tilde{\nabla}_n$-nilpotent.

Extremely strong Artinian conjecture. For every $n\in\mathbb{N}$, the subcategory $\mathcal{F}^{\omega-ad(n)}$ is stable under quotients.

Noetherian nilpotence conjecture. An object of finite type in $\mathcal{F}$ is noetherian of type $n-1$ if and only if it is $\tilde{\nabla}_n$-nilpotent.

Three-types conjecture. There are only three types of simple objects in $\mathcal{F}_{quad}$: those in the image of $\iota$, those in the image of $\kappa$, and those which are sub…

Let $\mathcal{F}_{surj}$ be the surjective-functor category, let $ev_n$ denote evaluation at the relevant object of dimension $n$, and let $\varpi(S)$ be the Powell functor associa…

Let $\mathcal{F}_{surj}$ be the category of functors associated with surjections. A category is co-locally artinian when its finitely cogenerated objects are artinian. The co-local…

Let $\Bbbk$ be the underlying field and let $P_{\Bbbk^{\oplus n}}$ denote the standard projective functor associated with $\Bbbk^{\oplus n}$. Powell's strengthened artinian conject…

Let $\mathcal{F}_{\mathcal{G}r,n}$ be the category of Grassmannian functors of level $n$, let $\omega_n$ be the associated functor to $\mathcal{F}$, and let “simple noetherian of t…

Let $\mathcal{F}_{\mathcal{G}r,n}$ be the category of Grassmannian functors of level $n$, let $\omega_n$ be the exact functor to $\mathcal{F}$, and let $G^f_0$ denote the Grothendi…

Let $\mathcal{F}$ be the functor category, and call a functor omega-adapted of height at most $n$ when it has the property defined in the source for height $n$. The extremely stron…

Let $\mathcal{F}_{\mathcal{G}r,n}$ be the category of Grassmannian functors of level $n$, let $\mathcal{F}^{lf}_{\mathcal{G}r,n}$ be its full subcategory of locally finite objects,…

Let $\Bbbk$ be the underlying field and let $\mathcal{F}(\Bbbk)$ denote the category of functors under consideration. A category is locally noetherian when every finitely generated…

Let $\mathcal{X}$ and $\mathcal{Y}$ be essentially small idempotent complete additive categories. Write $\operatorname{Mod}-\mathcal{X}$ for the category of modules over…

Soit $K$ un corps de caractéristique $p>0$, soit $A$ un anneau, et soit $\mathcal{F}^{df}(A,K)$ la catégorie des foncteurs de la catégorie considérée à valeurs dans des espaces vec…

Soit $A$ un $p$-anneau fini, c'est-à-dire un anneau fini dont la caractéristique est une puissance de $p$, et soit $K$ un corps tel que $\mathrm{car}(K)\neq p$. Pour un foncteur…

Varieties-of-functors conjecture. It may be possible to develop a theory of varieties of functors in which such classes of functors are described by small categories of generators…

Let $\Gamma$ and $\mathcal{E}$ be the conjugate pair of small categories considered in the paper, and let $\mathcal{C}$ be a stable model category satisfying the authors' technical…

Let $H$ be a nondegenerate $F2$-quadratic space and let $D$ be a degenerate quadratic space. For a quadratic space $V$, write $iso_V$ for the corresponding isotropic functor,…