Categories of FI type: A unified approach to generalizing representation stability and character polynomials

We give sufficient conditions on a category C to admit such structure via the notion of categories of FI type. This class of categories includes the examples listed above, and extends further to new types of categories such as the categorical power FIm, whose modules encode sequences of representations of m-fold products of symmetric groups. The theory is applied in [9] to give homological and arithmetic stability theorems for various moduli spaces, e.g. the moduli space of degree n rational maps P1→Pm.