We develop a functor calculus for functors from FI, the category of finite sets and injections, to an arbitrary stable presentable infinity-category, which we call FI-objects. We introduce a class of FI-objects, those we call analytic, which are determined by their Taylor towers. We classify the homogeneous layers of Taylor towers and show that the Taylor coefficients we thus obtain admit natural transformations which, subject to a Tate vanishing condition, recover those Taylor towers. Finally, we show that in the rational setting, our FI-calculus is an elaboration of the phenomenon of representation stability, and we give an explicit representation-theoretic dictionary between our Taylor coefficients and representation stable FI-modules.
