CONTINUOUS QUIVERS OF TYPE <i>A</i> (III) EMBEDDINGS OF CLUSTER THEORIES

Abstract We continue the work started in parts (I) and (II) of this series. In paper, we classify which continuous quivers type A are derived equivalent. Next, define new ${\mathcal {C}(A_{{\mathbb {R}},S})}$ , call weak cluster category. It is a triangulated category, it does not have structure but has weaker notion “cluster theory.” show that original category [15] localization theories to be appropriate groupoids structures satisfy conditions for theories. describe relationship between different theories: some obtained from structures. The mutation appears (but structures) next paper [20].