Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim} \ Z_n$ and $\underset{\rightarrow}{\lim} \ \Omega Z_n$

Question

Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell.edu/~hatcher/AT/AT.pdf, section 4.F, last lines) it's stated that the natural map $$\underset{\rightarrow}{\lim} \ \Omega Z_n \rightarrow \Omega \underset{\rightarrow}{\lim} \ Z_n$$ is a weak homotopy equivalence (the map is given by the universal property of the direct limit); I also recall that the direct limit of a sequence of CW-complexes is the mapping telescope. I'm trying to prove this fact but I don't know how to proceed; in particular, I don't know how to relate the homotopy groups of the mapping telescope to the homotopy groups of the spaces $Z_n$. There is a relation for the homology groups, namely $$H_i(\underset{\rightarrow}{\lim}\ Z_n)\simeq\underset{\rightarrow}{\lim}H_i(Z_n)$$ but I don't know how this can help.

Answer 1

The key intuition is that the loop space is the space of maps from $S^1$, and that such mapping spaces commute with direct limits up to weak equivalence because $S^1$ is a finite complex.

More formally, the claim follows from cellularity for relative complexes: any map $(X,A)\to Y$ of relative complexes which is cellular on $A$ is homotopic rel $A$ to a map which is cellular on $X$. Thus every map from a finite relative complex $(K,K')$ into $\varinjlim X_i$ factors up to homotopy rel $K'$ through a finite subcomplex, and thus (perhaps up to another relative homotopy) through some $X_i$.

In particular, this applies when $(K,K')=(S^k,*)$, implying surjectivity of the desired map on $\pi_k$. For injectivity, we use $(K,K')=(S^k\wedge I_+,S^k\vee S^k)$. The factorization result here implies that if maps $f,g:S^k\to X_i$ become homotopic in $\varinjlim X_i$, they become homotopic already in some $X_j$, which gives the result.

Answer 2

We have $\Omega = \operatorname{Map}_*(S^1, -)$ and $S^1$ is a compact topological space. Therefore, $\Omega$ commutes with directed colimits given by closed $T_1$-inclusions (such as a mapping telescope of CW complexes).

In more words, wlog we may assume $\{Z_n\}$ is a sequence of CW inclusions by passing to the mapping telescope. Since $S^1$ is compact, its image under a map $S^1 \to \varinjlim Z_n$ can't escape more than finitely many $Z_n$, and so every map $S^1 \to \varinjlim Z_n$ factors through some initial part $Z_m$, thus determining an element in $\Omega Z_m$ and in $\varinjlim \Omega Z_m$. This provides an inverse to the natural map $\varinjlim \Omega Z_n \to \Omega \varinjlim Z_n$.