Timeline for Why can we take the colimit over the category of elements?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 5 at 20:38 | vote | accept | themathandlanguagetutor | ||
| Mar 5 at 19:14 | comment | added | R. van Dobben de Bruyn | The category of finite étale covers of a locally Noetherian scheme is essentially small; in fact, even the category of finitely presented $X$-schemes is essentially small for any scheme $X$. This is an exercise in unwinding what 'finitely presented' means. | |
| Mar 5 at 12:00 | answer | added | Giacomo | timeline score: 5 | |
| Mar 5 at 3:26 | history | edited | themathandlanguagetutor | CC BY-SA 4.0 |
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| Mar 5 at 3:20 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Mar 5 at 3:16 | comment | added | themathandlanguagetutor | And it seems to me like the proof of injectivity relies on the description of the colimit as a quotient of the disjoint union. | |
| Mar 5 at 3:13 | comment | added | themathandlanguagetutor | @OmarAntolín-Camarena If I'm understanding the proof correctly, Murre uses the universal property of the colimit to produce a function $\operatorname*{colim}_{(S, \tau) \in \mathcal I^\mathrm{op}} \to F(X)$, then argues that this is both a surjection and an injection. | |
| Mar 5 at 3:10 | comment | added | themathandlanguagetutor | @DavidRoberts those are the ones! Though the copy I'm looking at is typewritten T_T | |
| Mar 5 at 2:56 | comment | added | Maxime Ramzi | It depends what exactly you mean by pro-representable. As Omar points out presumably the existence of the colimit follows from.the arguments. But if the indexing set is large (and that can happen if $\mathcal C$ is not assumed small from the start) then the representing "pro-object" is a large pro-object and as such not usually called a pro-object | |
| Mar 5 at 2:48 | comment | added | Dmitri Pavlov | Murre proves that the resulting cocone over the indicated diagram of sets is a colimit cocone. In particular, his argument proves that the colimit exists. | |
| Mar 5 at 2:22 | comment | added | David Roberts♦ | If $\mathcal{I}$ is a full subcategory of $\mathcal{E}$, and the latter is known to be a set (i.e. small, in modern terminology), then you are done. So the trick is to see that $\mathcal{E}$ is (essentially) small. | |
| Mar 5 at 2:18 | comment | added | Omar Antolín-Camarena | I'm a little confused: if you say Murre proves that the colimit you are asking about is F(X), that also shows it exists. Or does Murre somehow only show the colimit is F(X) under the assumption that the colimit exists? That's possible but sounds unlikely. | |
| Mar 5 at 2:14 | comment | added | David Roberts♦ | These notes mathweb.tifr.res.in/sites/default/files/publications/ln/… ? | |
| Mar 4 at 21:12 | history | edited | themathandlanguagetutor | CC BY-SA 4.0 |
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| S Mar 4 at 20:49 | review | First questions | |||
| Mar 4 at 21:15 | |||||
| S Mar 4 at 20:49 | history | asked | themathandlanguagetutor | CC BY-SA 4.0 |