2
$\begingroup$

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.

$\endgroup$

1 Answer 1

2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.