Let $k$ be a field, $\mathsf{Vect}$ the category of finite dimensional vector spaces, and $\mathsf{C} = Fun(\mathsf{Vect},\mathsf{Vect})$ the abelian category of pointed endofunctors (sending $0$ to $0$). The question has 2 parts, one for characteristic $0$ fields and the other for characteristic $p>0$.

Assume$char(k)=0$

**Definition**: An endofunctor $F \in \mathsf{C}$ is called **polynomial** if it can be expressed as a sum

$$F(V) = \bigoplus_{n \in \mathbb{N}}P(n) \otimes_{k[S_n]}V^{\otimes n}$$

Where $P(n)$ are $k$-linear representations of the symmetric group s.t. $P(n)=0$ for $n \gg 0$.

Questions:

1.Is there acharacterizationof thepolynomial functorsamong all endofunctors?

2.What's anexampleof anon-polynomial functorin $\mathsf{C}$?

3.Is $\mathsf{C}$ asemi-simplecategory? If so is there anexplicit descriptionof it? (hopefully identifying it with something built out of categories of representations).

Assume$char(k) = p \gt 0$

I think that in this case $\mathsf{C}$ will probably never be semi-simple (my intuition being that the exact sequence $0 \to \bigwedge^2 \to \otimes^{2} \to S^2 \to 0$ doesn't seem to be split in general) and so there looks to be some **interesting** (in my opinion) **homological algebra** going on here.

Question: Is $\mathsf{C}$ well understood in this case? By which I mean that there's a full list of all isomorphism classes and allExt groupscan be calculated in principle.

And finally, **where can I read more about this topic?**