Hyperplane arrangements are a classical meeting point of topology, combinatorics and representation theory. Generalizing to arrangements of linear subspaces of arbitrary codimension, the theory becomes much more complicated. However, a useful observation is that many natural sequences of arrangements seem to be defined using a finite amount of data.

In this talk I will describe a notion of 'finitely generation' for diagrams of arrangements, unifying the treatment of known examples. Such collections turn out to exhibit strong forms of stability, both in their combinatorics and in their cohomology representation. This structure makes the appearance of 'representation stability' transparent and opens the door to generalizations.