Let $\mathbf{FinVec}$ denote the category of finite-dimensional real vector spaces and linear maps, let $F\colon\mathbf{FinVec}\rightarrow\mathbf{FinVec}$ be a functor. For each pair of objects $V,W$, the Hom-set $\operatorname{Hom}(V,W)$ is itself a finite-dimensional vector space and we equip it with the unique structure of a topological space/of a smooth manifold that is compatible with its vector space structure. The composition maps are then continuous/smooth. In this way, we view $\mathbf{FinVec}$ as enriched over the category $\mathbf{Top}$/$\mathbf{Diff}$ respectively. It then makes sense to ask whether the functor $F$ is enriched over $\mathbf{Top}$/$\mathbf{Diff}$. That is just to ask whether the induced map $F\colon\operatorname{Hom}(V,W)\rightarrow\operatorname{Hom}(FV,FW)$ is continuous/smooth for all $V,W$.
(Irrelevant to the rest of the question, but the reason this is a useful notion is that it is the natural hypothesis assuring that the functor induces another functor on the category of topological/smooth vector bundles, applying the original functor fiberwise.)
In the case $V=W$, by functoriality, we can restrict $F$ to a group homomorphism $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$. If the functor is continuous/smooth, so is this map. Note that $\operatorname{GL}(V)$ is a Lie group. It is a general result that a measurable group homomorphism between Lie groups is automatically smooth. This begs the following questions:
- What's an argument for the existence of functors that aren't continuous? Can we construct one? Given the above, it either happens that the restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$ are continuous and continuity fails elsewhere, or the induced maps will be non-measurable, so that the construction will necessarily have to involve choice to some extent.
- Are there continuous functors that aren't smooth? Given the above, a continuous functor will have smooth restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$, but smoothness may fail elsewhere.
Best Answer
Reinstate Monica gives a construction in the comments to the question I linked in the comments (I can't close as a duplicate because that question has no upvoted or accepted answers): first, the category $\text{FinVec}(\mathbb{C})$ of finite-dimensional complex vector spaces has discontinuous endofunctors given by pulling back along discontinuous automorphisms $\mathbb{C} \to \mathbb{C}$. Second, these give rise to discontinuous endofunctors of $\text{FinVect}(\mathbb{R})$ given by complexifying, applying one of these automorphisms, then restricting back down to a real vector space of twice the dimension.
I suspect every continuous endofunctor is smooth, and in fact I suspect every continuous endofunctor is a (possibly infinite) sum of Schur functors, and hence even polynomial (by which I mean the induced maps $\text{Hom}(V, W) \to \text{Hom}(F(V), F(W))$ are polynomial; there are a few other possible definitions and I don't think they agree in general). The "possibly infinite" is to handle examples like the exterior algebra $V \mapsto \wedge^{\bullet}(V)$.