Timeline for Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 8, 2019 at 14:01 | vote | accept | Manuel Bärenz | ||
| Jan 6, 2019 at 15:44 | comment | added | Nik Weaver | I really like your formulation of "design principles". Well put. | |
| Jan 6, 2019 at 15:15 | comment | added | Peter LeFanu Lumsdaine | @MikeMiller — absolutely, if you don’t quotient the smooth/pw-linear maps then you only get an equivalence of ∞-categories. I understood the original question to be implicitly speaking of the homotopy categories, which is where you do classically get an equivalence of 1-categories. | |
| Jan 6, 2019 at 15:04 | comment | added | mme | As an addition to your excellent answer more specific to this problem: it should probably be noted that if there is an equivalence of categories here, it is an equivalence of $\infty$-groupoids of PL homeomorphisms vs diffeomorphisms. It is unlikely that the corresponding 1-categories agree (asking for automorphism groups to be isomorphic, as opposed to mapping class groups). You similarly wouldn't expect an equivalence of categories between all continuous maps and all smooth maps, but would between the corresponding $(\infty,1)$-categories. | |
| Jan 6, 2019 at 14:52 | history | answered | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |