Timeline for Is a smallness condition necessary in the Tannaka reconstruction theorem?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2021 at 8:01 | comment | added | Bugs Bunny | No, it is not. Take ${\mathcal C}$ to be vector space. Then $Nat(U_A,U_A)$ is typically not a vector space. | |
| Nov 26, 2021 at 11:47 | vote | accept | yohei ohta | ||
| Nov 26, 2021 at 9:55 | comment | added | yohei ohta | 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, 2021 at 8:25 | comment | added | Bugs Bunny | It is a set because your category is "presentable". It has a generator ${}_AA$, a value on which determines a natural endomorphism. | |
| Nov 26, 2021 at 0:41 | comment | added | yohei ohta | 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 25, 2021 at 12:11 | history | answered | Bugs Bunny | CC BY-SA 4.0 |