The connection between May's axioms for a triangulated tensor product and Happel's description of the derived category of the quiver D₄

In an important recent paper 12, May gave an axiomatic description of the properties of triangulated categories with a symmetric tensor product. The main point of the current article is that there are two other results in the literature which can be used to shed considerable light on May's work. The first is a construction of Verdier's, which appeared in Beilinson, Bernstein and Deligne's 4, Prop. 1.1.11, pp. 24-25]. The second and more important is the beautiful work of Happel, in 9, which can be used to better organise May's axioms.