Pushout-product of anodyne extensions is again anodyne

Question

Currently, I'm reading Simplicial Homotopy Theory by Jardine and Goerss and I got stuck in the proof of the theorem about pushout-products of anodyne extensions (corollary 4.6 in the book). Namely, I have no idea how to prove that for a fixed embedding $i\colon Y\subseteq X$ a class $$\{ u\colon K \rightarrowtail L \colon \text{ the pushout-product of } u \text{ and } i \text{ is anodyne} \}$$ is saturated. It seems to be obvious, but a straightforward attempt to prove that, for example, this class is closed under pushouts got me nowhere. Any hint will be appreciated.

Answer 1

So the class of morphisms that you want to verify is saturated is given by all those morphisms $K'\to L'$ such that

$$ f:(K' \times X ) \cup (L'\times Y) \to (L'\times X) $$ is anodyne for an arbitrary inclusion $Y \hookrightarrow X$. As saturated includes in particular that the class of morphisms has to be pushout-stable, you may consider firstly a pushout of $K'\to L'$ along a morphism, say $K'\to K''$, and denote the pushout by $K''\to L''$. Now you have to show that $K''\to L''$ is a member of the initially defined class and to do so, you check

$$ g : (K'' \times X) \cup (L''\times Y) \to (L'' \times X) $$ is anodyne. This follows from the fact that the class of anodyne morphisms is saturated and the fact that $g$ is the pushout of $f$ along

$$ (K'\times X) \cup (L' \times Y) \to (K'' \times X) \cup (L''\times Y). $$ Keep in mind that you need to proof that this diagram is cocartesian in the category of simplicial sets, which is a functor category (namely those functors from $\Delta^\mathrm{opp}$ to $\mathbf{Sets}$), so effectively you only need to show this in $\mathbf{Sets}$. Proceed similarly with the other properties that are needed to show saturatedness.