33 problems

Swiss cheese conjecture. There is a universal $n+1$-algebra $\mathrm{Hoch}(A)$ which acts on $A$.

A $d$-algebra is an algebraic structure whose operations are governed by the homology of configuration spaces of punctured $d$-dimensional discs; in particular, a $1$-algebra is as…

Let $\mathscr{P}$ be an operad, let $\mathbf{D}(\Bbbk)$ be the derived category over a field $\Bbbk$, and consider the enhanced bar-cobar adjunction … Here the left-hand category c…

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 $\lambda$ be the decategorification functor from Definition, let $\mathsf{Meas}$ be the ambient category of measurable structures, and let $\mathsf{2Vect}^{BC}$ denote the 2-ca…

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…

Johnson–Freyd's conjecture. Every rigid algebra in a fusion 2-category over $\mathbb{C}$ is automatically separable.

Let $d\geq 2$. An $E_{(d-1)}$ algebra has Hochschild cohomology equipped with the corresponding higher Hochschild structure. Higher Deligne's conjecture. The $E_{(d-1)}$ Hochschild…

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

Markl–Shnider adaptation conjecture. The construction of Markl and Shnider should adapt to the multiplihedra to define such a morphism. The proposed construction would provide an e…

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…

Invertibility conjecture. The algebra $\mathcal{A}$ is invertible if, and only if, it is $(n+1)$-dualizable and the canonical maps

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…

Fix a compact Lie group $G$, a monotone symplectic manifold $X$, and a Hamiltonian $G$-action with equivariant moment map $4\mu\colon X\to\mathfrak{g}^*4$. Let…

Let $A$ be an $E_{d-1}$-algebra, and let its deformation complex be the complex governing deformations of the $E_{d-1}$-algebra structure. Higher Deligne conjecture. The deformatio…