Timeline for Why do we care about $(\infty,2)$-categories?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 14, 2021 at 19:23 | comment | added | Dmitri Pavlov | “As for 2-categories, one doesn't even have to motivate them since they're all over the place”: most of these 2-categories likely have (∞,2)-analogues with the same motivation. I'd have a hard time imagining a situation where such a homotopical analogue is not possible, but it would be helpful if some concrete examples were given by the OP. | |
| Aug 13, 2021 at 2:29 | answer | added | dhy | timeline score: 10 | |
| Aug 11, 2021 at 7:44 | comment | added | Konrad Waldorf | I have a kind of déjà-vu. 15 years ago we were challenged to explain why to step from 1-categories to 2-categories. Now we have progressed to an $\infty$-analogue :-) See mathoverflow.net/questions/82981/are-higher-categories-useful/… | |
| Aug 11, 2021 at 2:36 | comment | added | David Ben-Zvi | One more example (again not at all unrelated): monoidal categories come up in many contexts in math. Their derived version is most naturally studied from the point of view of $(\infty,2)$-categories (the Morita theory of monoidal categories forms such a beast, just like the Morita theory of algebras is one of the most natural sources of ordinary categories) | |
| Aug 11, 2021 at 2:35 | answer | added | Andrew Macpherson | timeline score: 19 | |
| Aug 11, 2021 at 2:34 | comment | added | David Ben-Zvi | The three examples - 2-categories of bordisms, 2-categories of spaces with correspondences, and 2-categories of categories -- come packaged together when we study Lagrangian topological field theories (in codim 2) which are functors from the first to the third that factor through the second. | |
| Aug 11, 2021 at 2:32 | comment | added | David Ben-Zvi | There are lots of [$\infty$-] categories we might care about in mainstream math, and performing operations on them often requires a formalism of [$\infty$-]2-categories.. One example (not unrelated to my favorite, the TFT one) is treating the functoriality of categories of sheaves under correspondences - this is the main point of the book of Gaitsgory-Rozenblyum. | |
| Aug 11, 2021 at 0:42 | comment | added | Sridhar Ramesh | Probably the main reason to care about them is because, as you already note, people care about $(\infty, 1)$-categories, and people care about $2$-categories, and it seems likely they have some clean unifying common generalization. The sharpest natural notion of which would be the $(\infty, 2)$-categories. | |
| Aug 10, 2021 at 21:25 | comment | added | Arun Debray | A related/relevant MO question: mathoverflow.net/questions/373079 | |
| Aug 10, 2021 at 18:06 | history | asked | Daniel Teixeira | CC BY-SA 4.0 |