6 problems

Let $G$ be a finite group. A $G$-transfer system is a partial order on the set of subgroups of $G$ that refines the closed finite-index subgroup relation and is closed under conjug…

Blumberg–Hill's equivalence conjecture. $N_\infty$-operads are equivalent to $G$-indexing systems.

Blumberg–Hill tensor product conjecture. Their Boardman–Vogt tensor product should correspond to the join:

Suppose $G$ is a group and $\mathscr{O}$ is an $N_\infty$ operad: a $G$-operad whose spaces are $a3$-free, whose fixed-point subspaces are empty or contractible, and which has fixe…

Tensor product conjecture. Subject to suitably cofibrancy conditions, ${\mathcal O}\otimes{\mathcal O}'$ is an $N_\infty$ operad and

Barratt–Eccles sub-operad conjecture. If ${\mathcal C}$ is an indexing system and ${\mathcal F}$ is the associated sequence of families, then $\mathbb O^{{\mathcal F}}$ is a sub-op…