28 problems

The paper concerns algebras over a prop in chain complexes of vector spaces, and considers analogous homotopical and model-categorical constructions for simplicial modules. Simplic…

Let ${\cal P}$ be a quadratic Koszul operad, and let ${\cal B}^*_{{\cal P}_\infty}$ be the differential graded operad of natural operations on the complex defining the operadic coh…

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…

Enhancement conjecture. The equivalence above should be an equivalence of $\mathcal{C}_{k+1}$-algebras. This is presented as a conjectural strengthening of the preceding theorem, a…

Let $R$ be a based (that is, augmented) algebra over $C_{\ast}\mathcal{C}_k$, where $k\geq 1$, and let its Quillen and Hochschild cohomologies be taken in the category of based…

Let ${\cal A}_{\mathfrak D}$ be the colored operad of ${\mathfrak D}$-diagrams of ${\cal A}$-algebras, let ${\cal R}_{\mathfrak D}=({\cal F}(F),d_{\mathfrak D})$ be a cofibrant res…

Let $S$ be a monoid, and let $\mathcal{F}$-PureMono be the class of pure monomorphisms of right $S$-acts whose Rees quotient is flat. A class of monomorphisms is cellularly generat…

Let ${\cal A}$ be the cooperad governing the OPE data, let $H$ be the state space regarded as an ${\cal A}$-algebra, and let $\mathfrak g$ be the $L_{\infty}$-algebra controlling f…

Let $(P,A)$ be a $\Sigma$-free substitude that is non-quasi-tame, meaning that \Pi_1(\text{{bf T}^{scriptstyle tt T+{1}}}) is not equivalent to a discrete groupoid. Let…

Higher mutual information conjecture. An analogue of the probabilistic framework in which higher mutual information quantities belong to the homotopical algebra of cocycles of high…

The paper discusses the formality theorem for Lie algebroids and the use of the twisting procedure. Extension conjecture. The authors conjecture that the preceding observation imme…

Let $G$ be a finite group and let $\mathfrak{p}$ be a homogeneous prime ideal of $H^*(BG,k)$, where $k$ is a field of characteristic $p$. In the stable module category of $G$, let…

Let $\mathfrak{U}$ be a half-plane web associated to data $({\cal T}^{(\mu)},Z_\mu,\underline{R},\underline{\beta})$, where ${\cal T}^{(\mu)}$ are the theories, $Z_\mu$ are their c…

Let the theories ${\cal T}^{(\mu)}$ and Interfaces $\mathfrak{I}^{(\mu,\nu)}$ be as defined in the section, and let $\underline{\rho}(\mathfrak{W})[\underline r_a]$ denote the ampl…

Let $\mathfrak{W}$ be a web of interfaces, let $\underline{R}^{\rm intfc}$ be the direct sum of the vertex complexes over all fans of theories, and let…

Let $\mathfrak{W}$ be a web whose vertices are cyclic junctions of Interfaces between theories, and let $\underline{R}_{\mu_1,\dots,\mu_n}$ be the chain complex obtained by taking…

Let $\mathfrak{g}$ be a Lie algebra and consider the canonical map … The left-hand complex, shifted by one, is interpreted as the complex of coderivations of the standard Chevalley…

Let $\mathcal{O}$ be an operad and let $\zeta\in MC(\mathcal{O}^*)$ be a Maurer–Cartan element. The complex … is equipped with a $\mathcal{G}_\infty$-algebra structure. Generalized…

Strong commutative monoid axiom conjecture. Under these hypotheses, the strong commutative monoid axiom holds in $\mathcal{M}$. This conjecture is motivated by the corresponding re…

Let $A$ be an $4\mathcal E_n$ algebra, and let $B A$ denote its bar construction, regarded as an $4\mathcal E_{n-1}$ Hopf algebra. Let $\Omega B A$ be the cobar construction applie…

Let $A$ be an associative algebra, and in the cyclic case a cyclic unital associative algebra. Its Hochschild cochain complex has Hochschild cohomology groups carrying a (BV) Gerst…

Let a Topological Vertex Operator Algebra have homology groups equipped with Lian and Zuckerman's BV-algebra structure. Lian–Zuckerman's conjecture. This BV-algebra structure lifts…

Let $\mathscr{M}$ be an abelian $n$-fold monoidal category with weak compatibility between its exact and monoidal structures, and let $e$ be its unit element. Generalized Deligne c…

Explicit resolution conjecture. For the associative operad, the resolution from Theorem MAIN has a deformation complex isomorphic to the Gerstenhaber–Schack complex of the diagram.

Let $d\geq 2$. A $d$-algebra is an algebra over the homology operad $H_{-\bullet}(E_d)$ of the operad $E_d$ of little $d$-cubes. Consider the pair consisting of a $d$-algebra and i…