  In a paper by Holm and Jorgensen, the cluster category $\mathscr{D}$ of type
$A_\infty$, with Auslander-Reiten quiver $\mathbb{Z} A_\infty$, is introduced.
Slices in the Auslander-Reiten quiver of $\mathscr{D}$ give rise to direct
systems; the homotopy colimit of such direct systems can be computed and these
"Pr\"ufer objects" can be adjoined to form a larger category. It is this larger
category, $\overline{\mathscr{D}},$ which is the main object of study in this
paper. We show that $\overline{\mathscr{D}}$ inherits a nice geometrical
structure from $\mathscr{D}$; "arcs" between non-neighbouring integers on the
number line correspond to indecomposable objects, and in the case of
$\overline{\mathscr{D}}$ we also have arcs to infinity which correspond to the
Pr\"ufer objects. During the course of this paper, we show that
$\overline{\mathscr{D}}$ is triangulated, compute homs, investigate the
geometric model, and we conclude by computing the cluster tilting subcategories
of $\overline{\mathscr{D}}$.
