  We generalize the construction of reflection functors from classical
representation theory of quivers to arbitrary small categories with freely
attached sinks or sources. These reflection morphisms are shown to induce
equivalences between the corresponding representation theories with values in
arbitrary stable homotopy theories, including representations over fields,
rings or schemes as well as differential-graded and spectral representations.
  Specializing to representations over a field and to specific shapes, this
recovers derived equivalences of Happel for finite, acyclic quivers. However,
even over a field our main result leads to new derived equivalences for example
for not necessarily finite or acyclic quivers.
  The results obtained here rely on a careful analysis of the compatibility of
gluing constructions for small categories with homotopy Kan extensions and
homotopical epimorphisms, as well as on a study of the combinatorics of
amalgamations of categories.
