# Is a smallness condition necessary in the Tannaka reconstruction theorem?

$$\newcommand{\dmod}{\text{-}\mathrm{mod}}$$ Let $$A$$ be a finite-dimensional $$k$$-algebra, $$A\dmod$$ be a category of finite-dimensional A-modules and $$\mathrm{U}_A:A\dmod \to \textbf{Vect}_k$$ be a forgetful functor. We can reconstruct $$A$$ as $$\mathrm{End}(\mathrm{U}_A)$$ by using Tannaka reconstruction thorem.

Question : Does the claim hold even if the assumption of "finite-dimensional" is excluded?

Yes, of course. You still have a natural homomorphism $$A\rightarrow END(U_A)$$. Since $${}_AA$$ is a free $$A$$-module, an endomorphism $$x\in END(U_A)$$ is determined by its value $$x_A$$ on $${}_AA$$. This proves that the natural homomorphism is an isomorphism: $$x_A \in End (A_{End_{{}_AA}})= End (A_{{A}})=A.$$
• Is $\mathrm{End}(U_A)$ a set? If F and G are functors, I think $\mathrm{Nat}(F,G)$ is not a set in general. Nov 26 at 0:41
• It is a set because your category is "presentable". It has a generator ${}_AA$, a value on which determines a natural endomorphism. Nov 26 at 8:25
• Thx! I would like to ask another related question. I think Tannaka reconstruction theorem holds for an arbitary complete closed monoidal category $\mathcal{C}$. Let A be a monoid in $\mathcal{C}$ ,$A\text{-}\mathrm{Mod}$ be a category of left A-modules in $\mathcal{C}$ and $U_A:A\text{-}\mathrm{Mod} \to \mathcal{C}$ be a forgetful functor. Is $\mathrm{Nat}(U_A,U_A)$ a object of $\mathcal{C}$? Nov 26 at 9:55
• No, it is not. Take ${\mathcal C}$ to be vector space. Then $Nat(U_A,U_A)$ is typically not a vector space. Nov 29 at 8:01