  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.
