FI-calculus and representation stability

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