23 problems

Let $H$ be a connected Hopf algebra of finite Gelfand–Kirillov dimension, with antipode $S$. The Nakayama automorphism conjecture. The Nakayama automorphism of the coalgebra $H$ is…

Let $H$ be a co-Frobenius Hopf algebra, meaning that it has a nonzero left integral (equivalently, any of the equivalent conditions characterizing co-Frobenius Hopf algebras). Its…

Let $K$ be an algebraically closed field and let $C$ be a $K$-coalgebra. The coalgebra $C$ is of tame comodule type if its finite-dimensional indecomposable comodules are parametri…

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…

Inverse-limit comparison conjecture. The map above is an isomorphism of the cohomology groups.

Comparison conjecture. There is an isomorphism of homology theories

Conjecture. There is an isomorphism of coalgebras

Let an $L$-molecule be an entanglement of several codipterous coalgebras and/or anti-codipterous coalgebras constructed from coassociative coalgebras via channel maps, and let an o…

Let $\mathsf{T}_{\mathsf{INC}}$ denote the tangent endofunctor on $\mathsf{INC}(\mathcal{C})$, and suppose either preceding existence theorem applies, so that the universal Lincs c…

Let $k$) be a field of characteristic $p$, and consider the $k$-chains functor from spaces to $[?]$-coalgebras, where Mandell's theorem concerns the cochains functor on $p$-complet…

Let $H$ be a non-cosemisimple Hopf algebra over $\Bbbk$ with the dual Chevalley property. Let $\mathcal{M}$ denote the set of simple subcoalgebras of $H$, and let $\mathcal{P}$ be…

Let $D$ be a finite dimensional subcoalgebra of a coalgebra $C$, and let $\underline{d}$ be a dimension vector. Brall–Thrall conjecture. If $D$ is of infinite corepresentation type…

Let $T\colon n\mapsto n+1$ be a Poisson map admitting a coalgebra symmetry $(\mathfrak{A},\Delta)$. Let the generators of the coalgebra symmetry satisfy the closure relation referr…

Let $D$ be a simply connected conilpotent cocommutative coalgebra, in the setting of the Dold–Kan correspondence between simplicial $\bmoduli$-modules and nonnegative chain complex…

Generalized fundamental theorem of coalgebras. If $ckappa$ is an uncountable regular cardinal, then the fundamental theorem of coalgebras should extend in the sense proposed in the…

Let $C$ be an $H$-module coalgebra and let $A$ be an $H$-module algebra, with Hopf-cyclic cohomology defined using suitable coefficients. General Hopf-cyclic pairing conjecture. Th…

Hereditary cofree-coalgebra conjecture. The cofree coalgebra $\mathbb K\langle x_1,\dots,x_n\rangle^0$ is hereditary, and hence its comodule category is completely described by the…

Simson's infinite tame-wild dichotomy conjecture. Every coalgebra $C$ is either tame or wild, but not both.

Let $H$ be a co-Frobenius Hopf algebra, equivalently, a Hopf algebra with a non-zero integral. Andruskiewitsch–Dascalescu conjecture. The coradical filtration of $H$ is finite. Thi…

Let $R$ be a commutative ring, let $\mathcal{Q}$ be a cooperad, and write $\mathcal{Q}\text{-}\operatorname{Coalg}_{R}$ for the category of $\mathcal{Q}$-coalgebras. A twisting mor…

Let $\mathcal C$ be a locally presentable $\mathbb K$-linear category. A set of objects is spanning if it detects morphisms, and an object is tiny if its Hom functor preserves all…

Let $\mathbf{Grp}$ be the category of groups, let $\mathbf{Zinb}^c$ denote coZinbiel coalgebras, and let $\mathbf{Com}^c$ denote cocommutative coalgebras. For a group $G$, write…

The conjecture. If $C$ has a left torsion rational functor, then $C$ is $\mathcal{F}$-Noetherian.