What is the use of homotopy pushouts in Lurie's HTT 2.1.4.10?

Question

I have trouble understanding the usage of homotopy pushouts in the proof of proposition 2.1.4.10 in Lurie's Higher Topos Theory. My trouble is the way he shows that for a map $j\colon S\to S'$, the induced functor $j_!\colon (\mathcal{Set}_\Delta)_{/S}\to(\mathcal{Set}_\Delta)_{/S'}$ preserves weak equivalences in the covariant model structure.

Given a covariant equivalence $f\colon X\to Y$ (over $S$), I want to show that $j_!f$ is a covariant equivalence as well. This seems to be the relevant diagram. The top horizontal arrow is induced by $f$ and hence by assumption an equivalence of simplicial categories. The bottom horizontal arrow is $j_!f$. From what I understand, the diagram Lurie draws (one of the backsides of my diagram) is a pushout. This follows from the pasting laws for pushouts and the facts that $\mathfrak{C}[-]$ preserves colimits. He then states that $\mathfrak{C}[-]$ preserves monomorphisms and hence (I believe by A.2.4.4) he deduces that it must be a homotopy pushout as well. According to Lurie we are done.

Here are my questions:

1. Why does this imply that the map $j_!f$ (the bottom horizontal map in my diagram) is an equivalence?
2. Is my assertion correct that he uses A.2.4.4 to deduce that the square is a homotopy pushout? I don't really see the relevant cofibration.
3. In a similar vein he mentions in the proof of 2.1.4.7 (2) that a square is a homotopy pushout to conclude that the bottom arrow in a sqaure is an equivalence. This is equally confusing to me: Do homotopy pushouts preserve weak equivalences in general?

Any hint would be helpful, so thanks in advance.

Answer 1

1. Suppose $f: X \to Y$ is a morphism in $(\mathcal{S}et_{\Delta})_{/S}$ and $j: S \to S'$. If $f$ is a covariant equivalence, then $\mathfrak{C}[X^\triangleleft \sqcup_X S] \to \mathfrak{C}[Y^\triangleleft \sqcup_Y S]$ is a categorical equivalence. Applying the functor $\mathfrak{C}[S'] \sqcup_{\mathfrak{C}[S]} -$ to this equivalence, we see that $$\mathfrak{C}[S'] \sqcup_{\mathfrak{C}[S]} \mathfrak{C}[X^\triangleleft \sqcup_X S] \to \mathfrak{C}[S'] \sqcup_{\mathfrak{C}[S]} \mathfrak{C}[Y^\triangleleft \sqcup_Y S]$$ is also a categorical equivalence. But we can identify these pushouts using Lurie's claim: this is just the morphism $\mathfrak{C}[X^\triangleleft \sqcup_X S'] \to \mathfrak{C}[Y^\triangleleft \sqcup_Y S']$. So $j_! f$ is also a covariant equivalence.

2. Sure. The morphism $S \to X^\triangleleft \sqcup_X S$ is a monomorphism of simplicial sets (since it is a pushout of the monomorphism $X \to X^\triangleleft$). As will be explained by Lurie, $\mathfrak{C}[-]$ converts monomorphisms to cofibrations, so $\mathfrak{C}[S] \to \mathfrak{C}[X^\triangleleft \sqcup_X S]$ is a cofibration of simplicial categories, and you are in a setting to apply proposition A.2.4.4.

3. Yes - the idea is that in good cases you compute homotopy pushouts by replacing one of the morphisms with a cofibration and computing the ordinary pushout, and weak equivalences are preserved by pushouts, again in good cases. (Here, "good cases" should mean something like left proper, but don't quote me on this.) Anyway, if $i: X \to X'$ is a cofibration, then you can show that $X^\triangleleft \sqcup_X S \to (X')^\triangleleft \sqcup_{X'} S$ is a monomorphism, so that again by Jacob's claim $\mathfrak{C}[X^\triangleleft \sqcup_X S] \to \mathfrak{C}[(X')^\triangleleft \sqcup_{X'} S]$ is a cofibration. So, as a pushout of a weak equivalence along a cofibration, the morphism $\mathfrak{C}[(X')^\triangleleft \sqcup_{X'} S] \to \mathfrak{C}[(Y')^\triangleleft \sqcup_{Y'} S]$ is also a weak equivalence.