Timeline for Motivation of the fundamental theorem of covering spaces
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2022 at 13:24 | comment | added | Will Sawin | @user481980 1) If the category of $G$-sets is equivalent to the category of finite étale covers. 2) It's a property of the pair of categories together with the pullback functor. One also doesn't have to prove this purely categorically, one can use properties of profinite groups. | |
| May 11, 2022 at 13:15 | comment | added | user481980 | you not only have to show that "connected" is a categorical property, but that the whole "that every finite étale cover of Y that is connected pulls back to a connected cover of X" is a purely categorical property of the category of finite étale covers. | |
| May 11, 2022 at 13:13 | comment | added | user481980 | "if this holds for a group G" - if what holds for a group G? | |
| May 11, 2022 at 12:55 | comment | added | Will Sawin | @user481980 "Connected" Is a categorical property in this context because it means "not the coproduct of two nontrivial finite étale covers" since coproduct = disjoint union. | |
| May 11, 2022 at 12:54 | comment | added | Will Sawin | @user481980 Yes, they're related. If this holds for a group $G$, then $G$ maps to the automorphism group of every finite étale Galois cover, which are compatible with the maps between finite étale Galois covers, and satisfy a universal property, making $G$ the inverse limit of the automorphism groups of the finite étale Galois covers. This summary is very rough - we need to fix a base point for it to work. | |
| May 11, 2022 at 12:44 | comment | added | user481980 | "then the corresponding property for étale covers (i.e. that every finite étale cover of Y that is connected pulls back to a connected cover of X)" why is that property the corresponding property? in general, if we have an equivalence of categories we can only translate back and forth between categorical properties of these categories (which can be formulated just using objects, morphisms, equality, and composition). But your property uses "connected". | |
| May 11, 2022 at 12:42 | comment | added | user481980 | "one can show that there exists a unique profinite group satisfying that theorem" is that related to the statement that universal properties describe an object up to iso? | |
| May 9, 2022 at 17:57 | history | answered | Will Sawin | CC BY-SA 4.0 |