Joint with: Kiyoshi Igusa and Job Rock

Continuous cluster categories were introduced starting with representations of the real line as a generalization of representations of the quiver An. We change the construction to allow for different orientations of the real line and obtain the previous model as a localization of the new model. We introduce continuous clusters as generalizations of the original clusters and give some information on their structure. We also introduce a generalization of mutation, called contin- uous mutation, that allows one to work with all geodesic laminations of the hyperbolic plane.

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