Timeline for Lifting ring homomorphism of Grothendieck rings to functor of semisimple categories
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 6, 2019 at 16:33 | comment | added | Noah Snyder | Ah great, yes that's much simpler. For some reason I was only thinking about autoequivalences. You can also modify my first family of examples to make it simpler in the same way, just take two different 3-cocycles and the identity map on the fusion ring. (Side note for readers who aren't Victor, in addition to Tambara-Yamagami's proof that $\mathrm{Rep}(D_8)$ and $\mathrm{Rep}(Q_8)$ are distinct, Victor gave a nice argument that they have different numbers of fiber functors and so are distinct.) | |
| Aug 6, 2019 at 16:08 | comment | added | Victor Ostrik | Here is a related (and perhaps easier) example: the categories $\text{Rep}(D_8)$ and $\text{Rep}(Q_8)$ have isomorphic Grothendieck rings (with an isomorphism sending classes of simple objects to classes of simple objects), both categories are symmetric and not monoidally equivalent (which was proved by Tambara-Yamagami). | |
| Aug 5, 2019 at 16:13 | vote | accept | Steven Sam | ||
| Aug 5, 2019 at 16:06 | comment | added | Noah Snyder | Everyone when they first learn about strictification imagines that it's making a smaller category, but actually it's making a much bigger one. You're essentially adding new objects which are formal tensor products of your old ones. So if $1_g$ denotes the 1-dimensional vector space in degree g, in the skeletal version $1_g \otimes 1_h = 1_{gh}$ while in the strict version there's an extra new object $1_g 1_h$ which is isomorphic but not equal to $1_{gh}$. You can still see the associator by comparing the two isomorphisms $1_x 1_y 1_z \rightarrow 1_{xyz}$. | |
| Aug 5, 2019 at 14:56 | comment | added | Noah Snyder | Edited to upgrade my deleted comment on a symmetric example to the main text since I think I have an argument that shows it works now. | |
| Aug 5, 2019 at 14:55 | history | edited | Noah Snyder | CC BY-SA 4.0 |
added 1515 characters in body
|
| Aug 5, 2019 at 4:43 | comment | added | Noah Snyder | You can’t make that example both strict and skeletal at the same time. If you make it skeletal then it’s easy to see what the associator is, it’s just a 3-cocycle. If you make it strict, then you have to make the category much bigger and it’s harder to see what’s going on. | |
| Aug 5, 2019 at 4:25 | comment | added | Steven Sam | I'm probably being too naive: but my original thinking was to first replace the monoidal categories by their strict monoidal versions. So in your example when C=D, cycling the 1-dimensional irreps doesn't seem like it should create any problems. But my main problem is I don't have a good intuition for tihnking about coherence issues so it's hard to know where to look for inconsistencies. | |
| Aug 5, 2019 at 3:21 | history | answered | Noah Snyder | CC BY-SA 4.0 |