Timeline for Every locally presentable $\infty$-category can be presented by a proper model category
Current License: CC BY-SA 4.0
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| Nov 6, 2022 at 18:16 | comment | added | Simon Henry | @TimCampion it doesn't preserves pushout: the pushout taken in the original category has no reason to be fibrant - let alone be universal as an algebraicly fibrant object. Though in the nice situation where cofibrations are mono there is a nice description of pushout of cofibrations in algebraically fibrant object which only involves pushout of cofibration in the base category (a lot of them) which is why I think left properness is preserved. But the argument is involved enough that I'm hoping something simpler might be available... | |
| Nov 6, 2022 at 18:12 | comment | added | Tim Campion | It seems to me that the forgetful functor $U$, from algebraically-fibrant objects to the original category, should preserve pushouts? (maybe I am making a mistake here though). Granted this, since the weak equivalences are preserved and reflected by $U$, it becomes straightforward to deduce that if the original model category is left-proper, then the model category of algebraically-fibrant objects is also left proper. | |
| Oct 23, 2022 at 6:34 | history | edited | Mike Shulman | CC BY-SA 4.0 |
correct first link
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| Oct 22, 2022 at 18:46 | history | asked | Simon Henry | CC BY-SA 4.0 |