y_0$. See Definition \[def:cyclic\] for an explanation of inequalities.

If $D^1$ is the disk bounded by $S^1$, then we think of the diagonal $X$ as an isotopy class of non-selfintersecting curves in $D^1$ between the non-neighbouring vertices $x_0$ and $x_1$, see Figure \[fig:cluster tilting subcategory\]. Two diagonals cross if their representing curves intersect in the interior of $D^1$.

Starting from $\cZ$ and an algebraically closed field $k$, Igusa and Todorov in [@IT:cyclicposets sec. 2.4] constructed a cluster category $\cC( \cZ )$ of Dynkin type $A_{ \infty }$, which has a similar combinatorial model to that of $\cC( A_n )$. To wit, $\cC( \cZ )$ is a $k$-linear Hom-finite Krull–Schmidt $2$-Calabi–Yau triangulated category; the indecomposable objects are in bijection with the diagonals of $\cZ$, and non-vanishing $\Ext^1$ groups correspond to crossing diagonals. Further properties of $\cC( \cZ )$ are given in Section \[sec:IT\].

Cluster tilting subcategories of the cluster categories $\cC( \cZ )$ {#subsec:B}
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Our first main result is a classification of the cluster tilting subcategories of $\cC( \cZ )$ (see Definition \[def:cluster\_tilting\]). Cluster tilting subcategories of $\cC( A_n )$ correspond to triangulations of a finite polygon $P$, that is, maximal sets of pairwise non-crossing diagonals of $P$. By analogy, 