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We provide a graded and quantum version of the category of rooted clusteralgebras introduced by Assem, Dupont and Schiffler and show that every gradedquantum cluster algebra of infinite rank can be written as a colimit of gradedquantum cluster algebras of finite rank.  As an application, for each k we construct a graded quantum infiniteGrassmannian admitting a cluster algebra structure, extending an earlierconstruction of the authors for k=2.

Graded quantum cluster algebras of infinite rank as colimits