Timeline for Proof without sieves: Equivalent grothendieck topologies have the same sheaves
Current License: CC BY-SA 4.0
11 events
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| Aug 12, 2022 at 13:00 | history | edited | Muster Maxfrau | CC BY-SA 4.0 |
Adding a correction in terms of theory.
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| Aug 12, 2022 at 12:42 | vote | accept | Muster Maxfrau | ||
| Aug 12, 2022 at 8:15 | answer | added | Marc Hoyois | timeline score: 5 | |
| Aug 11, 2022 at 16:56 | comment | added | David Roberts♦ | The condition of being subordinate implies containment in the saturation, saturation preserves containment, and saturation is an idempotent operation. At least, my tired brain suggested this line of attack. | |
| Aug 11, 2022 at 16:45 | comment | added | მამუკა ჯიბლაძე | In Vistoli's notes there is the notion of saturated topology (Definition 2.52). I believe this is more or less the same as what others call topology. | |
| Aug 11, 2022 at 16:42 | comment | added | Mike Shulman | If you have access to it, section C2.1 of Sketches of an Elephant has a nice comprehensive discussion of pretopologies (or, rather, even weaker structures called "coverages") and their relation to "sifted" topologies (those defined in terms of sieves). | |
| Aug 11, 2022 at 16:19 | history | edited | Muster Maxfrau | CC BY-SA 4.0 |
added 22 characters in body
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| Aug 10, 2022 at 21:37 | history | edited | Muster Maxfrau | CC BY-SA 4.0 |
added 13 characters in body
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| Aug 10, 2022 at 21:30 | comment | added | Muster Maxfrau | Thank you Marc. That cleared some questions for me. First of all, yes I mean pretopologies. But then what does induced topology mean (I'm new with the notion of that topic :D)? If we have a Grothendieck Pretopologie, the induced topology is the Grothendieck topology with sieves? | |
| Aug 10, 2022 at 17:55 | comment | added | Marc Hoyois | Equivalent Grothendieck topologies are in fact equal, so they tautologically have the same sheaves. That is the point of using sieves in the definition of Grothendieck topology... Sieves tend to simplify proofs significantly. I assume your question is really about pretopologies, which are equivalent iff they induce the same topology. The simplest and most useful way to prove the statement is then to show that sheaves for a pretopology are the same as sheaves for the induced topology. | |
| Aug 10, 2022 at 14:53 | history | asked | Muster Maxfrau | CC BY-SA 4.0 |