map.

\[def-5\] The Thom spectrum $M=Mf$ of $f: b\to b{{gl_{1}}R}$ is the homotopy push-out $M$ of the diagram of ${E_{\infty}}$ spectra $$\label{eq:6}
\begin{CD}
{{\Sigma^{\infty}}_{+}}{\Omega^{\infty}}{{gl_{1}}R} @>>> R \\
@VVV @VVV \\
{{\Sigma^{\infty}}_{+}}{\Omega^{\infty}}p @>>> M,
\end{CD}$$ where the top map is the counit of the adjunction . Note that the spectrum underlying $M$ is the derived smash product $$\label{eq:60}
    M = {{\Sigma^{\infty}}_{+}}P {\wedge}^{L}_{{{\Sigma^{\infty}}_{+}}{{GL_{1}}R}} R,$$ generalizing the construction .

The adjunction between ${{\Sigma^{\infty}}_{+}}{\Omega^{\infty}}$ and ${gl_{1}}$ shows that, for any $E_\infty$ map $\varphi:R\to A$, we have a homotopy pull-back diagram of derived mapping spaces $$\begin{CD}
{(\mbox{${E_{\infty}}$ ring spectra})}  (M,A) @>>> {\mathscr{S}}(p,{{gl_{1}}A}) \\
@VVV @VVV \\
{(\mbox{${E_{\infty}}$ ring spectra})}  (R,A) @>>> {\mathscr{S}}({{gl_{1}}R},{{gl_{1}}A}),
\end{CD}$$ and comparing fibers over $\varphi$ gives the following.

\[t-th-Thom-iso\] There is a homotopy pull-back diagram $$\begin{CD}
{(\mbox{${E_{\infty}}$ $R$-$\mathrm{algebras}$})} (M,A) @>>> {\mathscr{S}}(p,{{gl_{1}}A})\\
@VVV @VVV \\
\{\varphi \} @>>> {\mathscr{S}}({{gl_{1}}R},{{gl_{1}}A}).
\end{CD}$$ That is, the space of $R$-algebra maps $M\to A$ is weakly equivalent to the space of lifts in the diagram .

An important feature of our construction is that it produces a Thom spectrum from any map $b \to b{{gl_{1}}R},$ not just from spherical fibrations. To make contact with the classical situation, let $S$ be the sphere spectrum, and suppose we are given 