Nakayama-type phenomena in higher Auslander--Reiten theory

This paper surveys recent contructions in higher Auslander–Reiten theory. We focus on those which, due to their combinatorial properties, can be regarded as higher dimensional analogues of path algebras of linearly oriented type $\mathbb{A}$ quivers. These include higher dimensional analogues of Nakayama algebras, of the mesh category of type $\mathbb{Z}{\mathbb{A}}_{\mathrm{\infty }}$ and the tubes, and of the triangulated category generated by an $m$-spherical object. For $m=2$, the latter category can be regarded as the higher cluster category of type ${\mathbb{A}}_{\mathrm{\infty }}$ whose cluster-tilting combinatorics are controlled by the triangulations of the cylic apeirotope.