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 $p>0$.



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> **Assume** $char(k)=0$
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**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 $n \gg 0$.




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> **Questions**: 
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> **1.** Is there a **characterization** of the **polynomial
>  functors** among all endofunctors?
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> **2.** What's an **example** of a **non-polynomial functor** in $\mathsf{C}$?
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> **3.** Is $\mathsf{C}$ a **semi-simple** category? If so is there an **explicit description** of it? (hopefully identifying it with something built out of categories of representations).
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> **Assume** $char(k) = p \gt 0$
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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.



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> **Question**: Is $\mathsf{C}$ well understood in this case? By which I mean that there's a full list of all isomorphism classes and all **Ext groups** can be calculated in principle. 
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And finally, **where can I read more about this topic?**



About the abelian category of endofunctors of $\mathsf{Vect}$
There are a couple of equivalent ways to characterise polynomial functors. One is to say that $F$ is a polynomial functor of degree $n$ if the function $$\hom(U, V)\to \hom(F(U), F(V))$$ is polynomial of degree n. A second, equivalent formulation goes via cross-effects: A functor is polynomial of degree $n$ if its $n+1$-th cross-effect vanishes. As far as I know, these definitions were first introduced in the classic paper 


*Eilenberg, Samuel; MacLane, Saunders*, [**On the groups $H(\Pi,n)$. II**](http://dx.doi.org/10.2307/1969702), Ann. Math. (2) 60, 49-139 (1954). [ZBL0055.41704](https://zbmath.org/?q=an:0055.41704).


In this paper the concept of cross-effect was introduced, and the equivalence of the two definitions proved.


In characteristic $0$, the category of polynomial functors is equivalent to the direct sum of representation categories of symmetric groups. In particular, it is semi-simple. This seems to be due to MacDonald


*Macdonald, I.G.*, [**Polynomial functors and wreath products**](http://dx.doi.org/10.1016/0022-4049(80)90128-0), J. Pure Appl. Algebra 18, 173-204 (1980). [ZBL0455.18002](https://zbmath.org/?q=an:0455.18002).


In positive characteristic, the category is not semi-simple. The following papers introduced the systematic study of functor categories between vector spaces over $\mathbb F\_p$, and in particular related them to modules over the Steenrod algebra 


*Henn, Hans-Werner; Lannes, Jean; Schwartz, Lionel*, [**The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects**](http://dx.doi.org/10.2307/2375065), Am. J. Math. 115, No.5, 1053-1106 (1993). [ZBL0805.55011](https://zbmath.org/?q=an:0805.55011).


*Kuhn, Nicholas J.*, [**Generic representations of the finite general linear groups and the Steenrod algebra. I, II, and III**](http://dx.doi.org/10.2307/2374932), Am. J. Math. 116, No.2, 327-360 (1994). [ZBL0813.20049](https://zbmath.org/?q=an:0813.20049).


There has been a lot of work on the homological algebra in this category, with striking applications in particular by Friedlander and Suslin to the homology of general linear group (**Edit**: for this application one needs the category of *strict* polynomial functors - a variant of the "naive" notion of polynomiality). However I don't think there is an effective procedure known for calculating the ext groups between general polynomial functors. For an up to date survey of the subject you may want to consult, for example, the following book


*Franjou, Vincent*, *Touze, Antoine* [**Lectures on functor homology.**](http://www.springer.com/la/book/9783319213040), , Proceedings of the conference on functor homology, Nantes, France, April 2012. Cham: Birkh\"auser/Springer (ISBN 978-3-319-21304-0/hbk; 978-3-319-21305-7/ebook). Progress in Mathematics 311, 7-39 (2015). [ZBL1338.18011](https://zbmath.org/?q=an:1338.18011).


Lastly let me mention that Kuhn also proved  that the some results about the category of polynomial functors between vector spaces over different characteristics is in fact semi-simple. See his comment below.


*Kuhn, Nicholas J.*, [**Generic representation theory of finite fields in nondescribing characteristic**](http://dx.doi.org/10.1016/j.aim.2014.12.012), Adv. Math. 272, 598-610 (2015). [ZBL1354.18001](https://zbmath.org/?q=an:1354.18001).


