An interesting categorification of formal power series

Let $\mathsf{N}$$\mathsf{N}$ be the groupoid such that the objects class are all the natural number and the morphisms are all the $S_n$$S_n$.

The functors from $\mathsf{N}$$\mathsf{N}$ to $\mathcal{C}$$\mathcal C$ give us an interesting way to consider the categorification of formal power series.

For example, let $\mathcal{C}$$\mathcal C$ be $\mathsf{FinSet}$$\mathsf{FinSet}$, let $F\in {\mathcal{C}}^{\mathsf{N}}$$F\in\mathcal C^\mathsf{N}$ be a functor with $F([n])=\{\tau \in P(P([n])):\tau \text{ is a topology}\}$$F([n])=\{\tau\in P(P([n])):\tau\text{ is a topology}\}$.

We could consider the decategorification as follows:

$$F⟼\sum _{n=0}^{\mathrm{\infty }}|F([n])|X^n$$

Or

$$F⟼\sum _{n=0}^{\mathrm{\infty }}|F([n])|{\displaystyle \frac{X^n}{n!}}$$

 

Or, since we have a natural $S_n$$S_n$ actions on $F([n])$$F([n])$, we could consider

$$F⟼\sum _{n=0}^{\mathrm{\infty }}|F([n]/S_n)|X^n\text{ and }F⟼\sum _{n=0}^{\mathrm{\infty }}|F([n]{)}^{S_n}|X^n$$

Notice that for two functors $F,G$$F,G$, we could consider $(F+G)([n])=F([n])+G([n])$$(F+G)([n])=F([n])+G([n])$

The decategorifications gives us

$$F+G⟼\sum _{n=0}^{\mathrm{\infty }}(|F([n])|+|G([n)]|)X^n$$

Also we could consider the convolution of functors $F\ast G([n])=\sum _{i+j=n}F([i])\times G([j])$$F* G([n])=\sum_{i+j=n} F([i])\times G([j])$

The decategorification gives us

$$F\ast G⟼\sum _{n=0}^{\mathrm{\infty }}\sum _{i+j=n}(F([i])\times G(j))$$