Timeline for Why is $1$ not a dense sub-site in a group with the trivial Grothendieck topology?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Aug 19, 2022 at 15:43 | answer | added | Simon Henry | timeline score: 16 | |
| Aug 19, 2022 at 15:16 | history | edited | Peter LeFanu Lumsdaine | CC BY-SA 4.0 |
fixed a couple of ambiguities in title
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| Aug 19, 2022 at 15:14 | vote | accept | Arshak Aivazian | ||
| Aug 19, 2022 at 15:07 | answer | added | Peter LeFanu Lumsdaine | timeline score: 16 | |
| Aug 19, 2022 at 14:59 | comment | added | Arshak Aivazian | And the article below says that Johnston's definition is wrong. This answers the question, thanks everyone, sorry for not thinking to look it up in the encyclopedia first. | |
| Aug 19, 2022 at 14:51 | comment | added | Arshak Aivazian | hmm, nlab has a different definition ncatlab.org/nlab/show/dense+sub-site | |
| Aug 19, 2022 at 14:36 | comment | added | Arshak Aivazian | In (ii) it is said for any morphism $f$ (with $\mathrm{cod}~f \in D$) to find a morphism $g$ such that (1) $fg \in D$ (2) the sieve generated by $g $ is a sieve from the site. Or am I reading wrong? If true, then $g := f^{-1}$ satisfies the condition. | |
| Aug 19, 2022 at 14:31 | comment | added | Peter LeFanu Lumsdaine | @MaximeRamzi: As written, it seems to me that (ii) is satisfied: the definition requires just that the given morphisms generate a covering sieve. In this case, there’s only one such morphism, but it’s an isomorphism, so it generates the maximal sieve. I agree with OP, this seems like a counterexample to the lemma as printed, and I would guess the error is exactly in the details of condition (ii). | |
| Aug 19, 2022 at 14:26 | comment | added | Peter LeFanu Lumsdaine | @ZhenLin: no — Johnstone specifically remarks, after the definition: “In practice, the Comparison Lemma is most often used for full subcategories, and many texts only define denseness in this case; however, the extra generality afforded by the definition we have given is occasionally useful — we shall see an instance of its use in 5.2.5 below.” | |
| Aug 19, 2022 at 14:18 | comment | added | Maxime Ramzi | Isn't (ii) not satisfied ? For f the identity of the single object, you need a covering sieve (i.e. all of G) such that the composite is always in D (i.e. trivial) | |
| Aug 19, 2022 at 13:25 | comment | added | Zhen Lin | I think the claim should probably be restricted to full subcategories. | |
| Aug 19, 2022 at 12:42 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
added 7 characters in body
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| Aug 19, 2022 at 12:37 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
added 5 characters in body
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| Aug 19, 2022 at 12:21 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |