Let $\textsf{FinVec}$ denote the category of finite-dimensional real vector spaces. For each vector space $V \in \textsf{FinVec}$, choose a vector space isomorphism $\varphi \colon V \to \mathbb{R}^{\dim V}$ and endow $V$ with the standard smooth structure from $\mathbb{R}^{\dim V}$ pulled back along $\varphi$. I think (correct me if I'm wrong!) that this smooth structure on $V$ is independent of the choice of $\varphi$ up to diffeomorphism.
Definition Let $n \in \mathbb{N} \cup \{\infty\}$ (where we use the convention that $\mathbb{N}$ includes $0$). A functor $F \colon \textsf{FinVec} \to \textsf{FinVec}$ is called $C^n$ iff $$f \mapsto F(f) \colon \operatorname{Hom}(V,W) \to \operatorname{Hom}(F(V), F(W))$$ is a $C^n$ function for all $V,W \in \textsf{FinVec}$ (where the smooth structures on the hom sets are the ones described above, as these hom sets are finite dimensional real vector spaces under pointwise addition & scalar multiplication).
Of course, any $C^n$ functor is also $C^m$ whenever $m \leq n$, and in particular $C^\infty$ functors are $C^n$ for all $n$. Examples of $C^\infty$ functors include:
- $\operatorname{Hom}(V,{-})$ for any $V \in \textsf{FinVec}$
- $V \otimes {-}$ for any $V \in \textsf{FinVec}$
- The exterior algebra functor
However, I've been unable to think of any examples of an endofunctor of $\textsf{FinVec}$ which is not $C^\infty$! So:
Does there exist an endofunctor of $\textsf{FinVec}$ which is not $C^\infty$?
If the answer is yes, then my follow-up questions are:
Does there exist an endofunctor of $\textsf{FinVec}$ which is not $C^0$? If $n,m \in \mathbb{N} \cup \{\infty\}$ with $m < n$, does there exist an endofunctor of $\textsf{FinVec}$ which is $C^m$ but not $C^n$?