96 problems

For each $k$, let $\mathsf{M}_k^{{||}}$ and $\mathsf{M}_k^{{\ddagger}}$ denote the parallel and series transition matrices, and let $\mathrm{Eul}_{k}(t)$ be the Eulerian polynomial…

For each $k$, let $N_k$ be the matrix appearing in the expression for the generating series $\mathcal{R}_k(t)$, and write $\pi_{N_k}(t)$ and $\chi_{N_k}(t)$ for its minimal and cha…

Let $\operatorname{\textit{Quilt}}$ be the operad of quilts, and let $H_k(\operatorname{Quilt})$ denote its homology in degree $k$. Acyclicity conjecture for the Quilt operad.…

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…

Deligne's conjecture. There exists a natural action of the operad $\mathsf{Chains}(C_2)$ on the Hochschild complex $\mathcal{C}^\bullet(\mathcal{A},\mathcal{A})$ for any associativ…

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…

Rubin's saturation conjecture. For every $n$ there exists some $f(n)$ such that, for any collection of distinct primes $p_1,\ldots,p_n\geqslant f(n)$ and any choice of…

Generalized pushout conjecture. There is a notion of pushout of ${\cal C}$-categories such that the loop space functor from spaces to ${\cal C}$-groupoids takes pushouts to pushout…

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…

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

Let ${\cal P}$ be a quadratic Koszul operad. Let ${\cal S}^*_{{\cal P}_\infty}$ be the suboperad of natural operations generated by the restricted class of operations generalizing…

Let ${\cal B}_{\it Lie}^*$ be the differential graded operad of natural operations on the Chevalley–Eilenberg complex of a Lie algebra with coefficients in itself, and let…

Bar-complex resolution conjecture. If $\mathcal{P}$ is a Koszul operad, then ${{\mathsf{B}}_{\bullet}^{\mathcal{P}}\!A}$ is a resolution of the left $A$-module…

Intrinsic-bracket conjecture. The homomorphism $I$ induces an isomorphism of operads

Let ${\mathcal L{\it ie}}$ be the Lie operad, and let ${\mathcal B}_{{\mathcal L{\it ie}}}$ be the operad of natural operations on the cohomology of Lie algebras. Lie-operad conjec…

Let ${\mathcal A{\it ss}}$ be the associative operad, and let ${\mathcal B}_{{\mathcal A{\it ss}}}$ be the operad of natural operations on the cohomology of associative algebras. L…

Gray-operad conjecture. There is a unique contractible $n$-operad $G_n$ such that, for every $k$ with $0\leq k\leq n-1$, \text{cal P}_k(G_n) is the free $k$-fold monoid operad.…

Binary-composition conjecture. Every weak $n$-category is weakly equivalent to an $A$-algebra.

Let $m,n$ be positive integers, let $\xi^m_n$ denote the generator indexed by $m$ inputs and $n$ outputs, and let $\partial$ be the perturbed differential in the minimal model of t…

For $n\ge 2$, let $MT(n)$ be the set of isomorphism classes of meta-trees with $n$ tails, and define a partial order $<$ on $MT(n)$ by declaring ${\bf T}<{\bf T}^{\prime}$ when the…

For $n\ge 2$, let $FM_2(n)$ be the Fulton–MacPherson space, let $MT(n)$ be the set of isomorphism classes of meta-trees with $n$ tails, and let $P$ be the dg-operad whose standard…

Stronger twisted coalgebra conjecture. For some model of the $\mathbb{E}_d$ operad, this zig-zag can be promoted to a zig-zag of equivalences