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I have a question about the proof of left anodyne between two simplicial sets over $S$, where $S$ is a simplicial set, is covariant equivalence. In the proof (HTT 2.1.4.6 or https://ncatlab.org/nlab/show/model+structure+for+left+fibrations, proposition 4.1), they formed the pushout diagram and then conclude the statement. However, I do not why this can conclude the result. I suppose this is because $\Lambda[n+1]_{i+1}\rightarrow \Delta[n+1]$ is a weak equivalence. Then why I also get the result for the coproduct on $S$?

This question is originally asked in stackexchange but has no reply. I have deleted that question and put it here.

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  • $\begingroup$ 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? $\endgroup$
    – Tim Campion
    Commented Apr 21, 2021 at 17:52
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    $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Commented Apr 21, 2021 at 17:52
  • $\begingroup$ @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. $\endgroup$
    – Peter Liu
    Commented Apr 21, 2021 at 18:17
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    $\begingroup$ 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. $\endgroup$
    – Tim Campion
    Commented Apr 21, 2021 at 18:36
  • $\begingroup$ @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. $\endgroup$
    – Peter Liu
    Commented Apr 21, 2021 at 19:06

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