Cluster structures for the $A_{\mathrm{\infty }}$ singularity

We study a category ${\mathcal{C}}_2$ of $\mathbb{Z}$-graded MCM modules over the $A_{\mathrm{\infty }}$ curve singularity and demonstrate it has infinite type $A$ cluster combinatorics. In particular, we show that this Frobenius category (or a suitable subcategory) is stably equivalent to the infinite type $A$ cluster categories of Holm-Jorgensen, Fisher and Paquette-Yildirim. As a consequence, ${\mathcal{C}}_2$ has cluster tilting subcategories modelled by certain triangulations of the (completed) $\mathrm{\infty }$-gon. We use the Frobenius structure to extend this further to consider maximal almost rigid subcategories, and show that these subcategories and their mutations exhibit the combinatorics of the completed $\mathrm{\infty }$-gon.