Timeline for Counter-example to the existence of left Bousfield localization of combinatorial model category
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 14, 2020 at 14:41 | answer | added | David White | timeline score: 5 | |
| Mar 13, 2019 at 23:19 | history | became hot network question | |||
| Mar 13, 2019 at 22:49 | vote | accept | Simon Henry | ||
| Mar 13, 2019 at 22:21 | comment | added | Simon Henry | @TimCampion : I thought it was in Hirschhorn's book, but he does it for "cellular" model categories instead of combinatorial. I just found the statement in Barwick "On left and right model categories and left and right bousfield localizations" as theorem 4.7. He attributed the results to J.Smith. I thought it was more "well known" than that. But maybe I missed a more classical reference. | |
| Mar 13, 2019 at 22:03 | comment | added | Tim Campion | I see that Casacuberta and Chorny showed all Bousfield localizations exist in a left proper, combinatorial, simplicial model category. Is the "simplicial" condition removed somewhere? | |
| Mar 13, 2019 at 22:00 | answer | added | Reid Barton | timeline score: 26 | |
| Mar 13, 2019 at 21:59 | comment | added | Tim Campion | Ah yes -- the statement that every orthogonality class is reflective is equivalent to weak Vopenka's principle -- this is 6.24 and 6.25 in Adamek and Rosicky. Example 6.25 is an example of an orthogonal subcategory in a locally presentable category which is not reflective (under the negation of weak Vopenka's principle), which I suppose answers your question in a rather artificial way. | |
| Mar 13, 2019 at 21:44 | comment | added | Simon Henry | Though I suspect one needs the negation of Vopenka's principle to get such an example ? | |
| Mar 13, 2019 at 21:32 | comment | added | Simon Henry | @TimCampion : good question. The left Bousfield localization at $S$ of a category where only iso are weak equivalence exists if and only if the subcategory of objects orthogonal to $S$ is reflective. it seems to me that this is not always the case when $S$ is a class, and that there might be known counter example to this ? but I haven't really thought about it. | |
| Mar 13, 2019 at 21:18 | comment | added | Tim Campion | Is there a standard example where $S$ is a class? | |
| Mar 13, 2019 at 21:08 | history | asked | Simon Henry | CC BY-SA 4.0 |