FI-calculus and representation stability

We introduce a functor calculus for functors $\mathsf{F}\mathsf{I}\to \mathcal{V}$, which we call $\mathsf{F}\mathsf{I}$-objects, for $\mathsf{F}\mathsf{I}$ the category of finite sets and injections and $\mathcal{V}$ a stable presentable $\mathrm{\infty }$-category. We show that $n$-homogeneous $\mathsf{F}\mathsf{I}$-objects are classified by representations of ${\mathfrak{S}}_n$ in $\mathcal{V}$, allowing us to associate “Taylor coefficients” to an $\mathsf{F}\mathsf{I}$-object. We show that these Taylor coefficients, in aggregate, themselves carry the structure of an $\mathsf{F}\mathsf{I}$-object, and we show that, up to the vanishing of certain Tate constructions, “analytic” $\mathsf{F}\mathsf{I}$-objects can be recovered from their $\mathsf{F}\mathsf{I}$-object of Taylor coefficients. We then establish a close relationship between our $\mathsf{F}\mathsf{I}$-calculus and the phenomenon of representation stability for $\mathsf{F}\mathsf{I}$-modules, suggesting that $\mathsf{F}\mathsf{I}$-calculus be understood as the extension of representation stability to the $\mathrm{\infty }$-categorical setting. In this context, we show how representation-theoretic information about a representation stable $\mathsf{F}\mathsf{I}$-module can be read off from its $\mathsf{F}\mathsf{I}$-module of Taylor coefficients.