  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.

Algebraic Topology;Algebraic Geometry;Category Theory;Representation Theory