Timeline for Homotopical categories, the 2-out-of-6 property, and saturation
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 25, 2015 at 7:57 | comment | added | Chris Schommer-Pries | (2) Weak equivalences are supposed to be inverted, but sometimes when you invert some arrows it forces other arrows to be inverted. Saturation is important since it tells you you know exactly which arrows get inverted. (1), 2-out-of-6 is something you can actually check sometimes. Saturated homotopical categories can still be a mess to deal with. It is nice to have the definition, but I think the more useful concept, at least if you ever want to compute anything, is what Barwick and Kan call a "partial model category". | |
| Sep 25, 2015 at 0:09 | answer | added | Tim Campion | timeline score: 3 | |
| Mar 27, 2015 at 11:19 | comment | added | user62675 | The reason one uses the 2 out of 6 property is because this implies that weak equivalences are saturated (however/also, see the discussion on page 12 here). As for the last two questions (which I think are answered briefly in the book linked to above), all model categories are saturated (see Remark 2.1.9 of Riehl's Categorical Homotopy Theory). | |
| Mar 27, 2015 at 10:24 | comment | added | Espen Nielsen | As for number 3, the class of weak equivalences of any model category (or closed model category in Quillen's original sense) is automatically saturated. | |
| Mar 27, 2015 at 10:00 | history | asked | Arrow | CC BY-SA 3.0 |