Timeline for Examples and counterexamples to Lack's coherence observation
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2022 at 6:09 | comment | added | Konrad Waldorf | E.g., what is the naturally occurring 2-category that is equivalent to the bicategory of algebras, bimodules, and intertwiners? | |
| May 11, 2022 at 6:08 | comment | added | Konrad Waldorf | I find this counter-intuitive. If the claim would be true, why working with bicategories? | |
| May 10, 2022 at 10:39 | comment | added | Maxime Ramzi | Well here it's natural in that the category of functors in question has a natural description : $C$-linear functors $C\to C$. This notion is interesting outside of coherence statements | |
| May 10, 2022 at 10:17 | comment | added | varkor | @MaximeRamzi: this is (a special case of) the proof of the coherence theorem for bicategories via Yoneda. I don't think it is reasonable to consider these 2-categories "naturally occurring", because otherwise Lack's observation becomes trivial. Of course, the observation isn't well-defined in the first place, so determining what "naturally occurring" even means is subtle, but I think it's fair to say that we should not expect any general coherence theorem to produce examples. | |
| May 10, 2022 at 9:55 | comment | added | Maxime Ramzi | I don't know enough to answer the question, but another example is (Joyal's ? or I forget who) the "Yoneda" proof of coherence for monoidal categories, namely that a monoidal category is equivalent to a strict one by mapping it to a certain category of endofunctors. Of course, by delooping, this gives an example of a bicategory being equivalent to a reasonably natural 2-category (namely "linear endofunctors") | |
| May 10, 2022 at 9:08 | comment | added | Peter LeFanu Lumsdaine | Very nice question indeed; I don’t know any answers, but look forward to seeing them! | |
| May 10, 2022 at 8:49 | history | asked | varkor | CC BY-SA 4.0 |