Tilting theory via stable homotopy theory

We show that certain tilting results for quivers are formal consequences of stability, and as such are part of a formal calculus available in any abstract stable homotopy theory. Thus these results are for example valid over arbitrary ground rings, for quasi-coherent modules on schemes, in the differential-graded context, in stable homotopy theory and also in the equivariant, motivic or parametrized variant thereof. In further work, we will continue developing this calculus and obtain additional abstract tilting results. Here, we also deduce an additional characterization of stability, based on Goodwillie’s strongly (co)cartesian n-cubes. As applications we construct abstract Auslander–Reiten translations and abstract Serre functors for the trivalent source and verify the relative fractionally Calabi–Yau property. This is used to offer a new perspective on May’s axioms for monoidal, triangulated categories.

Triangulated Categories

Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.