Timeline for Left anodyne is covariant equivalence
Current License: CC BY-SA 4.0
7 events
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| Apr 21, 2021 at 19:06 | comment | added | Peter Liu | @TimCampion Thanks a lot. At first I was thinking to use the category is left proper and try to use pushout of weak equivalence of cofibration is weak equivalence. I miss the map itself is acyclic cofibration. | |
| Apr 21, 2021 at 18:36 | comment | added | Tim Campion | Glad to help. I should add that the proof of the fact I mentioned goes something like this. A morphism $i$ is an acyclic cofibration if and only if it has the left lifting property with respect to all fibrations. So $i$ is an acyclic cofibration and if $j$ is a pushout of $i$, the thing to do is use lifts along $i$ and extend them via the pushout to get lifts along $j$. This works more generally for any so-called weak factorization system. | |
| Apr 21, 2021 at 18:17 | comment | added | Peter Liu | @TimCampion Thank you for adding the tag. Using theorem 3.3 will make things harder for me as I lack those kind of knowledge. But I quite agree with your second comment and this fully resolve my question. Thanks. | |
| Apr 21, 2021 at 17:52 | comment | added | Tim Campion | It also occurs to me that the confusion might be resolved by noting that in any model category, the pushout of an acyclic cofibration along any morphism is an acyclic cofibration (and that pushouts in slice categories are computed as in the underlying category). But I'm not sure I've understood the confusion. | |
| Apr 21, 2021 at 17:52 | comment | added | Tim Campion | I've added some tags, I hope you don't mind. It's generally considered good practice on MO to include at least one "top-level" tag (usually the ones corresponding to arxiv categories, like ct.category-theory or ag.algebraic-geometry), not the least because it provides increased visibility. The nlab gives two defining characterizations of the model structure -- Def 3.1 and Thm 3.3. Just to make sure we're on the same page, do you agree that from the description in Thm 3.3 this is obvious? | |
| Apr 21, 2021 at 17:49 | history | edited | Tim Campion |
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| Apr 21, 2021 at 16:00 | history | asked | Peter Liu | CC BY-SA 4.0 |