236 problems

Let $A$ be an algebra, $M$ an $A$-module, and let the deformation complex be the controlling differential graded Lie algebra associated with the deformation problem of $M$, such as…

Rubin's saturation conjecture. Cyclic groups are saturated as long as the prime factors of their order are sufficiently large.

Blumberg–Hill's classification conjecture. $N_\infty$-operads are completely classified by their associated $G$-transfer systems.

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

Let $C$ be the category appearing above, let $L{\mathsf{Sp}}$ be the corresponding localized category of spectra, and write $\partial^{BB}_\ast$ for the Goodwillie derivative funct…

Let ${\mathcal{SC}}_n$ be the Swiss cheese operad, let ${\mathcal E}_n$ be its second-color little-discs operad, and let…

Ault–Fiedorowicz conjecture. For every $k$-algebra $A$, there is an isomorphism

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…

Let $\mathbb{B}^{0}_C$ be Batanin's globular operad, whose algebras are his models of globular weak $\infty$-categories, and let $\Theta_{\mathbb{B}^{0}_C}$ be its associated theor…

Francis–Gaitsgory's conjecture. To obtain an equivalence of $\infty$-categories, it should suffice to restrict $B_{\mathcal{O}}$ to pro-nilpotent $\mathcal{O}$-algebras and…

Operadic realization conjecture. The expression $\left(1-\sum_{k\geq 1}p_k\right)^{-1}$ is Schur positive and can be expressed in terms of the free algebra over an operad. This con…

Model-structure conjecture. For a wide range of $A_{\infty}$-operads ${\cal C}$, the category of $\widehat{{\cal C}}$-precats admits a closed model structure with these cofibration…

Effective categorification conjecture. For any $A_{\infty}$-operad ${\cal C}$, there is an operation $Cat _{{\cal C}}$ from $\widehat{{\cal C}}$-precats to $\widehat{{\cal C}}$-cat…

Extension property converse. If every $(S,\underline D)$-diagram of ${\cal P}$-algebras can be extended to an $S$-diagram of ${\cal P}$-algebras, then the couple $(S,D)$ has the…

Let $V=\mathbb{R}^d$, let $\Phi$ be the bundle of local fields of a quantum field theory, and let $\Phi_0$ be its fiber at the origin. Let $G_d$ be the group of parallel translatio…

For $d\geq 0$, a $d$-algebra is an algebra over the operad $\mathsf{Chains}(C_d)$ in complexes, where $C_d$ is the little $d$-discs operad. The generalized Deligne conjecture. For…

An $A_{\infty}$-algebra is a homotopy associative algebra; its cohomological Hochschild complex is a differential graded Lie algebra. A $2$-algebra is an algebra over the operad…

Baez–Dolan's conjecture. The $n-k$-category of endomorphisms of the $k-1$-fold identity of $s$ in $\sigma^k$ is the Poincaré $n-k$-category $\Pi_{n-k}(X_k)$.

Let $\Gamma$ be the operad whose $k$-th space is the nerve of the category of permutations of $\{1,2,\dots,k\}$, and let $\Gamma^{(n)}$ be Smith's filtration, in which a simplex be…

The graphs defined by Kontsevich form a Feynman PROP $K$, and let $F$ denote the forest PROP introduced above. Freeness conjecture. The PROP $K$ is conjectured to be the free PROP…

Dendroidal sets form a category equipped with a notion of inner Kan complex; for dendroidal sets $X$, $Y$, and any dendroidal inner Kan complex $K$, write…

Let $Mag(V)$ be the free magmatic algebra on a vector space $V$, and let $NAP^c(V)$ denote the corresponding cofree $NAP^c$-coalgebra. The coalgebra map … is the canonical quotient…

A Lie-coalgebra–PostLie bialgebra is the conjectural bialgebra structure obtained by combining a Lie coalgebra with a PostLie algebra; a free PostLie algebra is the free algebra fo…

Let $T$ be a non-crossing tree of degree $n$, not necessarily based, viewed as an element of the operad of moulds $obreakoperatorname{Mould}$. Let $operatorname{Dend}$ denote the…

Aguiar–Loday's conjectures. The first conjecture concerns the dimensions of the $\mathbb{S}_n$-modules $\mathcal{Q}uad(n)$, and the second conjecture is that the operad…