16 problems

Let $\mathcal{C}$ be a Gray monoid. A braided and symmetric pseudomonoid additivity conjecture. Every braiding on $\mathcal{C}$ induces an equivalence … of 2-categories, and every…

Terminality conjecture. The object $\mathrm{Hoch}(A)$ is terminal in the homotopy category of Swiss cheese actions on $A$.

Let $p$ be the prime appearing in the definition of the DGAs $S_{2n}$, and let $l$ be a positive integer. The E-infinity refinement conjecture. For each $l>0$, the DGA $S_{2p^l-2}$…

Let $n\geq 1$ and let $A$ be an $\mathbb{E}_{n+2}$-algebra in $\operatorname{Sp}$. For a pointed space $X$, write $C^*(X;A)$ for the $A$-valued cochains on $X$, regarded as an augm…

The infinity-categorical structure conjecture. The simplicial sets $\mathrm{HOM}_{\mathsf{\Omega B As\text{-}Alg}}(A,B)_\bullet$ are $\infty$-categories.

Let $h_=h\Psi$ be the element defined in the last remark before the appendix of Dotsenko, Shadrin, and Vallette (2016). Let i_ and p_ be the $_\alpha$-morphisms whose composite…

Let $n$ be a nonnegative integer, let $\mathcal{S}$ denote the ambient symmetric monoidal category of spaces, and let $\mathcal{A}\in\operatorname{Alg}_n(\mathcal{S})$ be an $E_n$-…

Let a polyadic nonunital non-strict groupal category be a polyadic non-strict semigroupal category equipped with a querfunctor and quertors satisfying the groupal coherence conditi…

Let a polyadic ($n$-ary) tensor category be equipped with a unit object and unitors. Arity-reducibility conjecture. It should be arity-reducible to a binary category, with its $n$-…

$n$-ary coherence conjecture. If the $n$-ary associator $\scalebox{1.15}{\textsf{\upshape A}}^{\left(2n-1\right)\otimes}$ satisfies such $n$-ary coherence conditions that this isom…

Let $F\colon\mathcal A\to\mathcal B$ be a homological epimorphism of $A_\infty$-algebras, and let $\widetilde{\mathcal A}$ be an $E_n$-algebra enhancing $A$. The $E_n$-enhancement…

Let $\text{Fin}^{\text{op}}$ be the opposite category of finite sets, and let $\text{Burn}^{\text{eff}}$ denote the effective Burnside 2-category, or spans of finite sets. A cyclic…

Cayley–Dickson extension conjecture. The space $A'$ can be given the structure of a Cayley-Dickson imaginaroid; moreover, this structure is associative if $A$ is commutative.

Additivity conjecture. For every $k\geq 1$, the additivity isomorphism $E_{k+1}\simeq E_1\otimes^{\mathbf{L}}_{BV}E_k$ induces the stated map…

Let $\mathcal{R}\in\operatorname{CAlg}_{\operatorname{Fin}}$ and let $\mathcal{L}$ be a cyclic commutative algebra over $\mathcal{R}$, meaning a commutative $\operatorname{Fin}$-al…

Let $Q$ be a simply laced quiver, let $\mathcal{A}={\rm Rep}(Q)$ be the category of finite-dimensional representations over a finite field $\mathbb{F}_q$, and let $\mathcal{A}_0$ b…