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We call a 2-Calabi–Yau triangulated category a cluster category if its cluster-tilting subcategories form a cluster structure as defined in Buan et al. (Compos Math 145:1035–1079, 2009). In this paper, we show that the canonical orbit category of the bounded derived category of finite dimensional representations of a quiver without infinite paths of type \({\mathbb {A}}_\infty \) or \({\mathbb {A}}_\infty ^\infty \) is a cluster category. Moreover, for a cluster category of type \({\mathbb {A}}_\infty ^\infty ,\) we give a geometrical description of its cluster structure in terms of triangulations of an infinite strip with marked points in the plane.

Cluster categories of type $${\mathbb {A}}_\infty ^\infty $$ A ∞ ∞ and triangulations of the infinite strip