Timeline for What is the connection between direct/inverse image of set maps and direct/inverse image functors of sheaves?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 8, 2014 at 21:31 | answer | added | Yonatan Harpaz | timeline score: 2 | |
| Jan 16, 2014 at 12:08 | comment | added | Zhen Lin | I would say it's just an unfortunate coincidence of terminology. I don't know if anyone ever seriously thought about the right adjoint of $f^{-1} : \mathscr{P}(Y) \to \mathscr{P}(X)$ until Lawvere. | |
| Jan 16, 2014 at 10:26 | comment | added | KotelKanim | Well, I understand this. As I said myself, with these constructions, there is a connection but it is different (I wouldn't say the opposite) than what the names suggest. You claim that that is all there is, and I shouldn't expect anything else. That's absolutely fine :) | |
| Jan 16, 2014 at 10:08 | comment | added | Ketil Tveiten | Think about how you define the sections of direct/inverse image sheaves: $f_*\mathcal{F}(U)=\mathcal{F}(f^{-1}(U))$ etc.; the sections (think functions) of the direct image sheaf is sections on the inverse image of sets, so your constructions are connected, the correspondence just goes opposite to what the names suggest. | |
| Jan 16, 2014 at 10:00 | comment | added | KotelKanim | @KetilTveiten, I don't understand your comment. In both cases, there are two constructions; one covariant and the other contravariant. If you have an argument for why a connection between the two is not possible, please explain more. | |
| Jan 16, 2014 at 9:51 | comment | added | Ketil Tveiten | $X\mapsto\mathcal{P}(X)$ is covariant, $X\mapsto\mathcal{S}h(X)$ is contravariant, so you can't really expect them to be instances of the same general thing. | |
| Jan 16, 2014 at 9:35 | history | asked | KotelKanim | CC BY-SA 3.0 |