13 problems

Let the category of one-colored simplicial (wheeled) properads carry the model structure lifted from that on $\mathtt{sSet}^{\Sigma^{\operatorname{op}}\times \Sigma}$, and let the…

Classification conjecture. The morphism $\varphi_a$ is an isomorphism in weight $3$ if and only if $\mathcal{P}_a$ is Koszul. Moreover, these equivalent conditions hold if and only…

Conjecture. For every $a\in\mathbb{C}^4$, the following are equivalent: the properad $\mathcal{P}_a$ induces a confluent system; the morphism $\varphi_a$ is a bijection in weight…

Let $\Omega(\overline{\textsf N}^{fr})$ be the cobar construction of the properad of framed curves, let $\Omega(\textsf{Frob}^*)$ be the cobar construction of the dual Frobenius pr…

Properads are categories of operations with multiple inputs and outputs, equipped with composition by attaching a nonzero number of inputs of one operation to outputs of another; l…

A labelled cospan category is a symmetric monoidal category equipped with a symmetric monoidal functor to the category of cospans satisfying the pullback condition specified in the…

The properads $\mathcal{H}\mathit{olieb}$ and $\mathcal{L}\mathit{ieb}$ control homotopy Lie bialgebra structures, while their wheeled closures…

Let $\operatorname{sProperad}$ be the category of simplicially enriched properads, and let … be the properadic nerve functor, where $\varGamma$ is the graphical category. Properadi…

Folklore vanishing conjecture. The common cohomology group vanishes:

Koszulness conjecture. The properad ${\mathcal L}ie{\mathcal B}^{\diamondsuit}$ is Koszul.

Homology conjecture. The homology of the properad $\operatorname{\Omega}(\mathrm{co}\mathbf{Frob})$ is the involutive biLie properad.

Cobar resolution conjecture. The properad $\operatorname{\Omega}(\mathrm{co}\mathbf{Frob})$ gives a resolution of the Lie bialgebra properad.

Higher-homotopy properad conjecture. The pair $({\mathcal Ass\mathcal Bi}_\infty,\partial)$ is a one-coloured version of a certain $\mathbb{N}$-coloured properad describing…