\ \ The cluster category of Dynkin type $A_\infty$ is a ubiquitous object with interesting properties, some of which will be explained in this talk.
\\
\ \ Let us denote the category by $\mathcal{D}$. Then $\mathcal{D}$ is a 2-Calabi-Yau triangulated category which can be deﬁned in a standard way as an orbit category, but it is also the compact derived category $D^c(C^{∗}(S^2;k))$ of the singular cochain algebra $C^*(S^2;k)$ of the 2-sphere $S^{2}$. There is also a “universal” deﬁnition: $\mathcal{D}$ is the algebraic triangulated category generated by a 2-spherical object. It was proved by Keller, Yang, and Zhou that there is a unique such category.
\\
\ \ Just like cluster categories of ﬁnite quivers, $\mathcal{D}$ has many cluster tilting subcategories, with the crucial diﬀerence that in $\mathcal{D}$, the cluster tilting subcategories have inﬁnitely many indecomposable objects, so do not correspond to cluster tilting objects.
\\
\ \ The talk will show how the cluster tilting subcategories have a rich combinatorial
structure: They can be parametrised by “triangulations of the $\infty$-gon”. These are certain maximal collections of non-crossing arcs between non-neighbouring integers.
\\
\ \ This will be used to show how to obtain a subcategory of $\mathcal{D}$ which has all the properties of a cluster tilting subcategory, except that it is not functorially ﬁnite. There will also be remarks on how $\mathcal{D}$ generalises the situation from Dynkin type $A_n$ , and how triangulations of the $\infty$-gon are new and interesting combinatorial objects.