Timeline for Mac Lane strictness theorem and categorifiability of fusion rings
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 20, 2016 at 2:41 | vote | accept | Sebastien Palcoux | ||
| Jan 19, 2016 at 1:15 | comment | added | Sebastien Palcoux | @MarcelBischoff: Yes you're right, it is precisely remark 2.8.7 of this book. | |
| Jan 18, 2016 at 23:18 | comment | added | Marcel Bischoff | I think the confusion here is that one might think that for a strict category the 6j symbols are trivial and one could build a skeletal fusion category with trivial associater. The point is that this will in general not fulfill all the axioms. It is in general pretty difficult to check if a fusion ring has a categorification. To me it seems to be completely magical if a categorification exists, because the system of equations looks pretty overdetermined. That's why people strife to find a common source from which all fusion categories come from, conformal field theory is a big candidate. | |
| Jan 18, 2016 at 15:26 | answer | added | Dave Penneys | timeline score: 10 | |
| Jan 18, 2016 at 15:25 | comment | added | Todd Trimble | I can't quite tell what you're asking. If you're asking whether a fusion ring is the (Grothendieck ring) of a rig obtained by decategorifying a rigid strict monoidal category with coproducts over which the monoidal product distributes, even that doesn't seem trivially achievable (much less the added conditions you'd need for an actual strict fusion category). I just don't see how you'd manufacture such a structured category out of thin air, as it were. | |
| Jan 18, 2016 at 14:57 | comment | added | Sebastien Palcoux | @ToddTrimble: Is a fusion ring trivially categorifiable into a strict monoidal category, or is there a difficulty at this level? | |
| Jan 18, 2016 at 14:05 | comment | added | Todd Trimble | I must be very confused then. The pentagon condition is after all satisfied in a strict monoidal category. But I should probably bow out of the discussion since I'm not an expert on fusion categories and so am ill-placed to discuss where the real issue presumably lies. | |
| Jan 18, 2016 at 13:53 | comment | added | Sebastien Palcoux | @ToddTrimble: because for a strict monoidal category we don't need the pentagon axiom, so we don't need to know if the pentagon equation admits a solution (which is very hard a priori). | |
| Jan 18, 2016 at 13:45 | comment | added | Todd Trimble | If strict fusion category simply means a fusion category which is strict monoidal as a monoidal category, then I'm having trouble seeing why the answer to both questions isn't a trivial 'yes'. Could you amplify on why "too easy"? | |
| Jan 18, 2016 at 13:11 | history | asked | Sebastien Palcoux | CC BY-SA 3.0 |