Let $\mathcal{C}$ be a symmetric monoidal cocomplete category. Let $F:\mathbb{P}^{op}\rightarrow \mathcal{C}$ be a functor, where $\mathbb{P}$ denotes the permutation category. Such a functor is sometimes called a symmetric sequence or a $\mathbb{S}$-module or a $\mathcal{C}$-species. For any such symmetric sequence $F$ we can consider its associated analytic functor (also called Schur functors)
$$\hat{F}[X]:= \coprod_{n\geq 0} F[n]\otimes_{S_n} X^{\otimes n}.$$ These appear naturally as the monads associated to an operad as explained here. Why they are called analytic functors seems obvious: They look like categorifications (in an imprecise, intuitive sense) of formal power series. Now the ring of formal power series can be given a natural topology (several equivalent definitions exist, one is, for example, via the $(X)$-adic topology) and one can speak of convergence of formal power series.
This naturally raises the question:
- Is there a notion of convergence of analytic functors? Or does the analogy with analytic functions/formal power series end there?
In the realm of Goodwillies analytic $(\infty,1)$-functors, there seems to be a notion of convergence of analytic functors, but to my knowledge Goodwillie's analytic functors and those defined above are not related.