Timeline for Push-forward of a locally constant sheaf using two homotopic maps
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 3, 2023 at 0:03 | comment | added | Greg Friedman | @D.-C.Cisinski Thank you! | |
| Feb 2, 2023 at 22:55 | comment | added | D.-C. Cisinski | @asv This is an answer indeed. | |
| Feb 2, 2023 at 19:31 | comment | added | asv | @D.-C.Cisinski: It sounds like a final answer. Isn’t it? | |
| Feb 1, 2023 at 12:02 | comment | added | D.-C. Cisinski | @GregFriedman The pullback $f^*$ is defined for all maps $f$. Now, if two left adjoints are isomorphic, so are their right adjoints. Hence the right adjoints, whenever they exist, only depend on the homotopy type. In fact, all this depends on the weak homotopy type only, and up to a weak equivalence, any map is a Serre fibration, which implies that the right adjoint always exists. the point of being a proper submersion is that this right adjoint can be computed as the ordinary (derived) sheaf-theoretic push forward. | |
| Feb 1, 2023 at 1:02 | comment | added | Greg Friedman | The last statement of the preceding comment is not clear to me. If the adjointness property depends on $f$ being a submersion, why does it depend only on the homotopy type of $f$ when the homotopies can be through non-submersions? | |
| Jan 31, 2023 at 22:21 | comment | added | D.-C. Cisinski | Yes, it is true: locally constant sheaves only depend on the homotopy type of $X$, functorially with respect to pullbacks (i.e. the pullback functor along any map $f$ restricted to locally constant sheaves only depends on the homotopy class of $f$). When $f$ is a proper submersion, the functor $Rf_*$ is a right adjoint of the pullback and thus only depends on the homotopy type of $f$. | |
| Jan 31, 2023 at 14:21 | history | asked | asv | CC BY-SA 4.0 |