  We show that variants of the classical reflection functors from quiver
representation theory exist in any abstract stable homotopy theory, making them
available for example over arbitrary ground rings, for quasi-coherent modules
on schemes, in the differential-graded context, in stable homotopy theory as
well as in the equivariant, motivic, and parametrized variant thereof. As an
application of these equivalences we obtain abstract tilting results for trees
valid in all these situations, hence generalizing a result of Happel.
  The main tools introduced for the construction of these reflection functors
are homotopical epimorphisms of small categories and one-point extensions of
small categories, both of which are inspired by similar concepts in homological
algebra.
