Timeline for What determines a model structure?
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| May 11, 2016 at 21:48 | comment | added | Martin Frankland | A published reference (with credit to Joyal) for the fact that cofibrations and fibrant objects determine the model structure is Emily Riehl's Categorical Homotopy Theory, Theorem 15.3.1. | |
| Jun 18, 2015 at 10:06 | comment | added | Dmitri Pavlov | @TimCampion: Joyal's CatLab is still (and has always been) public, the link is ncatlab.org/joyalscatlab/published/… | |
| Jan 17, 2015 at 4:48 | comment | added | Tim Campion | I happened across this question again today, and noticed that my link is broken because Joyal's Catlab is no longer public. Anyway, there's a proof in paragraph 51.10 of Joyal's unpublished but widely disseminated notes on quasicategories, which he seems to have settled on calling "Notes on logoi" (even if this link breaks too, that name should remain easily searchable, I think!). | |
| May 19, 2012 at 8:52 | comment | added | Tim Campion | To clarify for anyone reading this: the second item in this response turns out to be wrong. As pointed out by other comments and answers, it's a theorem of Joyal that cofibrants + fibrations DO determine the model structure (although it seems to be more common to state the dual: cofibrations + fibrants determine...). An updated link to catlab proof: ncatlab.org/joyalscatlab/show/Model+categories#determination2 | |
| Jun 27, 2010 at 21:27 | vote | accept | roger123 | ||
| Jun 27, 2010 at 18:09 | comment | added | roger123 | Thank you for the example (This is point 4. not 3.). | |
| Jun 26, 2010 at 21:32 | comment | added | Charles Rezk | For 3, cofibrant objects and fibrant objects do not even determine weak equivalences. For instance, Cat(=small categories) admits model structures with all objects fibrant and cofibrant, with either equivalences or isomorphisms as the weak equivs. | |
| Jun 26, 2010 at 21:04 | history | answered | Mark Hovey | CC BY-SA 2.5 |