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This short note surveys the constructions of 3-Calabi--Yau triangulatedcategories with simple-minded collections due to Ginzburg andKontsevich--Soibelman and the constructions of 2-Calabi--Yau triangulatedcategories with cluster-tilting objects due toBuan--Marsh--Reineke--Reiten--Todorov and Amiot, and includes a discussion onthe normal form of 2-Calabi--Yau triangulated categories with cluster-tiltingobjects.

The interplay between 2-and-3-Calabi--Yau triangulated categories