Timeline for Can equivalences be strictified to isomorphisms?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2020 at 19:52 | vote | accept | Mike Shulman | ||
| Jan 30, 2018 at 18:48 | answer | added | Mike Shulman | timeline score: 3 | |
| Mar 27, 2012 at 22:59 | comment | added | David Roberts♦ | I think it is true that every Gray-group (a Gray-groupoid with one object) is equivalent to one arising from a 2-crossed module. One could see what the braiding becomes under this equivalence, but this isn't exactly an impossibility result, more like a conditional impossibility result (can't have your cake and eat it) | |
| Mar 27, 2012 at 17:39 | answer | added | Mike Shulman | timeline score: 6 | |
| Jun 8, 2010 at 20:29 | comment | added | Mike Shulman | Also, the notion of "1-cell isomorphism" doesn't make a whole lot of sense in a bicategory that's not a strict 2-category. Even identities in a bicategory are not usually isomorphisms! But one could ask about a braided monoidal strict-2-category which is weaker than a Gray-monoid, I guess. | |
| Jun 8, 2010 at 20:27 | comment | added | Mike Shulman | Well, I said "one way to make this question precise would be..." (-: I think the question is most interesting if we ask for some other semistrictness in addition, though, since we know that (in low dimensions at least) interchange can actually be turned into an identity as long as the unit constraints are weak enough. | |
| Jun 8, 2010 at 15:42 | comment | added | Peter LeFanu Lumsdaine | A priori, the final question you ask seems a shade stronger than what you started off with. There are other situations where it's know that you can make aspect A or aspect B of something strict, but not both at once; and it would seem conceivable that, say, every braided Gray-monoid could have its braiding strictified to an equivalence, but that this process might destroy the Gray-ness, giving just a braided monoidal bicategory, and then Gray-ifying it back might turn the braiding back into just an equivalence? Or does Gray-ification preserve the property of braiding being iso? | |
| Jun 5, 2010 at 23:07 | history | asked | Mike Shulman | CC BY-SA 2.5 |