Category Theory – Proposition A.2.6.15 in HTT

ct.category-theoryhigher-category-theorymodel-categories

This is a cross-post of a question in MSE.

I am reading Lurie's Higher Topos Theory and I need some help to understand a part of the proof of Proposition A.2.6.15. (A.2.6.13 in the published version) This proposition is used repeatedly in the book to construct various model structures.

In the proposition, we work with a locally presentable category $\mathbf{A}$, a class $W$ of morphisms in $\mathcal{C}$, and a set $C_0$ of morphisms in $\mathbf{A}$, subject to the following conditions:

  1. The class $W$ contains all isomorphisms, has the two out of three property, is close under filtered colimits, and has a set $W_0$ such that every morphism in $W$ is a filtered colimit of morphisms in $W_0$.

  2. Given cocartesian squares

$$\require{AMScd}
\begin{CD}
X @>>> X' @>{g}>>X''\\
@V{f}VV @VVV @VVV\\
Y @>>> Y' @>>{h}> Y'',
\end{CD}
$$

if $f\in C_0$ and $g\in W,$ then $h\in W$.

  1. Every map in $\mathbf{A}$ having the right lifting property with respect to $C_0$ lies in $W$.

We then want to show that the condition (2) remains valid if we repalce $C_0$ by the weakly saturated class generated by $C_0.$ Lurie proves this by arguing that the class $P$ of morphisms for which (2) remains true when $C_0$ is replaced by $P$ is weakly saturated, i.e., closed under pushouts, transfinite compositions, and retracts.

I understand that $P$ is closed under pushouts and transfinite compositions: Closure under pushout is obvious. Closure under transfinite transfinite composition is a consequence of the fact that $W$ is closed under filtered colimits. But I don't see why $P$ is closed under retracts. Can anyone help me figure this out? Any help is appreciated. Thanks in advance!

Best Answer

Retracts of weak equivalences are weak equivalences.

Now if $f'$ is a retract of $f$ and you start with such a diagram with $f'$ on the left, you can create a new diagram with $f$, the same $X', X''$ but new $Y',Y''$, determined by cocartesianness of the two squares.

The claim is that the original diagram is a retract of this new one. This follows essentially because retracts of cocartesian squares are cocartesian.

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