  We introduce a handy construction of cluster algebras of type
$\mathbb{A}_{\infty}$, we give a complete classification of the cluster
algebras arising from the infinity-gon, and finally we construct the category
of the diagonals of the infinity-gon and show that it is triangle-equivalent to
the infinite cluster category of type $\mathbb{A}_{\infty}$ described by Holm
and J{\o}rgensen.
