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

Abstract: Representation stability is a theory describing a way in which a sequence of representations of different groups is related, and essentially contains a finite amount of information. Starting with Church-Ellenberg-Farb's theory of $FI$-modules describing sequences of representations of the symmetric groups, we now have good theories for describing representations of other collections of groups such as finite general linear groups, classical Weyl groups, and Wreath products $S_n\wr G$ for a fixed finite group $G$. This paper attempts to uncover the mechanism that makes the various examples work, and offers an axiomatic approach that generates the essentials of such a theory: character polynomials and free modules that exhibit stabilization. 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 $FI^m$, whose modules encode sequences of representations of $m$-fold products of symmetric groups. The theory is applied in [Ga] to give homological and arithmetic stability theorems for various moduli spaces, e.g. the moduli space of degree n rational maps $P^1 \rightarrow P^m$.