Proof of Lemma 5.7 in Goerss & Jardine

Question

I am trying to understand the proof of Lemma $5.7$, chapter $4$ page $237$, in 'Simplicial Homotopy Theory', Goerss & Jardine.

The lemma:

Suppose that $X:I\rightarrow \textbf{sSet}$ is a simplicial set valued functor which is defined on a small category $I$. Suppose further that the induced simplicial set map $X(\alpha):X(i)\rightarrow X(j)$ is a weak equivalence for each morphism $\alpha:i\rightarrow j$ of the index category $I$. Then, for each object $j$ of $I$ $$X(j)\rightarrow \text{hocolim} X \rightarrow BI$$ is a homotopy fiber sequence (homotopy cartesian pullback in GJ).

I do not understand why it suffices to show that any $$\Lambda^n_k\xrightarrow{i} \Delta^n\xrightarrow{\sigma} B I$$ induces a weak equivalence $$\Lambda^n_k\times_{BI} \text{hocolim} X\xrightarrow{i_\ast} \Delta^n\times_{BI} \text{hocolim} X.$$

Argument in GJ: It follows from the small object argument because pulling back along $\text{hocolim} X \rightarrow BI$ preserves colimits.

I understand why pulling back preserves colimits, but I do not understand why this, together with the small object argument, gives the above simplification.

Answer 1

The point is to follow the small object argument through for the factorization of $j:*\to BI$ into a trivial cofibration followed by a fibration. So let $\Delta^n\to BI$ be such that its restriction to $\Lambda^n_k$ factors through $j$. Pushing out, we get a trivial cofibration $*\to U_1$ which is the first step of the small object argument. Pulling back this pushout square along $\pi:\mathrm{hocolim} X\to BI$, let $Q_1=\Delta^n\times_{BI}\mathrm{hocolim}X$, $P_1=U_1\times_{BI}\mathrm{hocolim} X$, and $R_1=\Lambda^n_k\times_{BI}\mathrm{hocolim} X.$ In this case, it happens that $R_1=\Lambda^n_k\times X(j)$, but that's of no real importance. Then what Goerss and Jardine remark is that we have a pushout square $P_1=Q_1\sqcup_{R_1} X(j)$. Since $i_*:R_1\to Q_1$ is a pullback of $\Lambda^n_k\to \Delta^n$, it is a cofibration; by assumption, it is a weak equivalence. Thus $X(j)\to P_1$, as a pushout of a trivial cofibration, is a trivial cofibration. Continuing in this way, pulled back from the trivial cofibrations $*\to U_1\to U_2\to ...$ we get trivial cofibrations $X(j)\to P_1\to P_2\to...$ Finally, transfinite composition gives the trivial cofibration $*\to U$ with its pullback, $X(j)\to P$, as claimed.