11 problems

Polygraph quotient conjecture. The homomorphism $\psi$ is an isomorphism.

Higher-dimensional polygraph conjecture. The coordinate ring of $Z(n,l)$ over $E=\mathbb{A}^d$ is a free $k[\mathbf{z}]$-module.

Self-dual coherent omega-equivalence conjecture. The augmented chain complex $C$ is the linearisation of a polygraph presenting a self-dual model of the coherent walking $omega$-eq…

Let $\mathcal{S}$ be the class of molecules with spherical boundary. Algebraic freeness conjecture. The class $\mathcal{S}$ of molecules with spherical boundary is algebraically fr…

Let $\mathbf{RAtom}_\textit{in}$ be the subcategory of inner regular atoms and let $\mathbf{RPol}:=\mathbf{PSh}_{}(\mathbf{RAtom}_\textit{in})$. The category $\mathbf{RPol}$ should…

Let $P$ be a directed complex. It is freely generating when $(\mathcal{M}o\ell_{}{P})^*$ admits the structure of a polygraph whose $n$-dimensional generators are the $n$-dimensiona…

Let $C$ denote the décalage construction on regular plexes described in the paper, and let a plex be a polygraph with the corresponding universal property. Plex-preservation conjec…

Let $\mathbf{CPol}$ be the category of constructible polygraphs and let $\mathbf{RCPol}$ be the category of representable constructible polygraphs and strong maps. For a constructi…

Regularity conjecture. Every regular directed complex should be freely generating, meaning that $(\mathcal{M}o\ell_{}{P})^*$ admits a polygraph structure whose $n$-dimensional gene…

Let $P$ be a directed complex. Say that $P$ is freely generating if $(\mathcal{M}o\ell_{}{P})^*$ admits the structure of a polygraph whose $n$-dimensional generators are the $n$-di…

Let $\mathbf{CPol}$ be the category of constructible polygraphs, with monoidal structures given by the lax Gray product $\scaleobj{0.75}{\boxtimes}$ and the join $\star$, and let…