Stabilization as a CW approximation (Q1300650)

Let \(S\) be the sphere spectrum and \({\mathcal M}_S\) denote the category of \(S\)-modules constructed by Elmendorf, Kriz, Mandell, and May. In this paper, it is shown that the map \(\Sigma^\infty:{\mathcal T}(X,Y)\to{\mathcal M}_S(\Sigma^\infty X,\Sigma^\infty Y)\) is a homeomorphism. Here \({\mathcal T}\) denotes the category of based compactly generated weak Hausdorff spaces. This implies that \(\Sigma^\infty\) is a full and faithful embedding of \({\mathcal T}\) in \({\mathcal M}_S\). In particular, the unstable homotopy class \([X,Y]\) is isomorphic to \(h{\mathcal M}_S(\Sigma^\infty X,\Sigma^\infty Y)\) and \(\Sigma^\infty X\) is not a stabilization of \(X\). This comes from the fact that all weak equivalences are not inverted. Suppose that \(X\) is homotopic to a CW complex. Then the homeomorphism \(\Sigma^\infty\) implies two theorems. One is that for a CW approximation \(\gamma:\Gamma \Sigma^\infty X\to\Sigma^\infty X\), the composite \({\mathcal T}(X, Y)@> \Sigma^\infty>>{\mathcal M}_S(\Sigma^\infty X,\Sigma^\infty Y)@> \gamma^*>>{\mathcal M}_S(\Gamma\Sigma^\infty X,\Sigma^\infty Y)\) induces the usual stabilization map \([X,Y]\to \{X,Y\}\). The other is that \(X\) is contractible, if \(\Sigma^\infty X\) is a cofibrant \(S\)-module. These are typical properties of the category \({\mathcal M}_S\) and seem to be ``well known to the experts''. The homeomorphism \(\Sigma^\infty\) is certified by showing that the top row in the following commutative diagram is an equalizer: \[ \begin{matrix} {\mathcal T}(X,Y) &\overset{\eta_*}\rightarrow &{\mathcal T}(X, \Omega^\infty\Sigma^\infty Y) & \overset{\Omega^\infty\widehat\xi_*}{\underset {\theta'} \rightrightarrows} &{\mathcal T}(X, \Omega^\infty F[{\mathcal L}(1), \Sigma^\infty Y))\\ \Sigma^\infty\downarrow && \downarrow\cong &&\downarrow \cong\\ {\mathcal M}_S(\Sigma^\infty X, \Sigma^\infty Y) &\overset\iota\rightarrow &{\mathcal S}{\mathcal U}(\Sigma^\infty X,\Sigma^\infty Y) & \overset{\widehat\xi_*} {\underset\theta\rightrightarrows} &{\mathcal S}{\mathcal U}(\Sigma^\infty X, F[{\mathcal L}(1),\Sigma^\infty Y))\end{matrix}. \]