  Church-Ellenberg-Farb used the language of FI-modules to prove that the
cohomology of certain sequences of hyperplane arrangements with S_n-actions
satisfies representation stability. Here we lift their results to the level of
the arrangements themselves, and define when a collection of arrangements is
"finitely generated". Using this notion we greatly widen the stability results
to:
  1) General linear subspace arrangements, not necessarily of hyperplanes.
  2) A wide class of group actions, replacing FI by a general category C.
  We show that the cohomology of such collections of arrangements satisfies a
strong form of representation stability, with many concrete applications.
  For this purpose we develop a theory of representation stability and
generalized character polynomials for wide classes of groups. We apply this
theory to get classical cohomological stability of quotients of linear subspace
arrangements with coefficients in certain constructible sheaves.
