On the tails of FI-modules

We study the end-behavior of integer-valued $\mathrm{FI}$-modules. Our first result describes the high degrees of an $\mathrm{FI}$-module in terms of newly defined tail invariants. Our main result provides an equivalence of categories between $\mathrm{FI}$-tails and finitely supported modules for a new category that we call $\mathrm{FJ}$. Objects of $\mathrm{FJ}$ are natural numbers, and morphisms are infinite series with summands drawn from certain modules of Lie brackets.