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

CONTINUOUS QUIVERS OF TYPE A (III) EMBEDDINGS OF CLUSTER THEORIES