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In this paper, we give a complete classification of cotorsion pairs in acluster category $\mathscr{C}$ of type $A^\infty_\infty$ via certainconfigurations of arcs, called $\tau$-compact Ptolemy diagrams, in an infinitestrip with marked points. As applications, we classify $t$-structures andfunctorially finite rigid subcategories in $\mathscr{C}$, respectively. We alsodeduce Liu-Paquette's classification of cluster tilting categories of$\mathscr{C}$ and Ng's classification of torsion pairs in the cluster categoryof type $A_\infty$.

Cotorsion pairs in cluster categories of type $A_{\infty}^{\infty}$