12 problems

Monotonicity conjecture. The global dimension is a monotone function on the poset of all transfer systems $\mathcal O$ for $G$, with minimal value $0$ attained only at the complete…

Let $p$ be prime and let $M$ be a $C_p$-Mackey functor. A projective resolution of $M$ is eventually $4$-periodic if its differentials repeat with period $4$ from some point onward…

Let $d$ be a nonnegative integer, and let ${\mathsf{OEpi}}_{\leq d}$ be the category of finite-dimensional inner product spaces of dimension at most $d$ and orthogonal epimorphisms…

Equivariant $SK$ realization conjecture. The $G$-Mackey functor formed by the equivariant $SK$-groups should be the $\pi_0$-Mackey functor of some genuine $G$-spectrum.

The categories of $\mathcal{O}_a$-Mackey functors carry a symmetric monoidal $\mathcal{O}_m$-Mackey functor structure. An $\mathcal{O}_m$-commutative monoid is a commutative monoid…

External Tambara conjecture. An $(\mathcal O_a,\mathcal O_m)$-Tambara functor is an $\mathcal O_m$-commutative monoid in the symmetric monoidal $\mathcal O_m$-Mackey functor of…

Coinduction preservation conjecture. Coinduction restricts to a functor

External Hoyer–Mazur conjecture. For any compatible pair of indexing categories $(\mathcal O_a,\mathcal O_m)$, there is an equivalence of categories between $\mathcal O_m$-commutat…

Norm functor conjecture. There is a norm functor

Let $\mathcal{M}=\mathcal{F}_s^{\leq n}$ be the category of finite sets and surjections of cardinality at most $n$, as in Example, and let $\mathbf{C}$ be a semiadditive presentabl…

Let $\mathcal{C}$ be a small category such that its finite coproduct completion $\operatorname{fam}\mathcal{C}$ is lextensive. Let $\operatorname{Aut}\mathcal{C}$ be the category o…

Equivariant generating hypothesis. The restriction of $\underline{\pi}_*^G(-)$ to the subcategory of finite $G$-spectra is faithful. Equivalently, if a map $f:X\to Y$ between finit…