**Background and notations:**
Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a *Thom space* $T_n(\xi)$, given by the cofiber:
$\require{AMScd}$
\begin{CD}
\ \xi @>>> B\\
@VVV @VVV\\
* @>>> T_n(\xi).
\end{CD}

This construction can be viewed as a (suitably homotopy invariant functor) from the slice category of spaces over $BGL_1S^{n-1}$ (which is the classifying space for $n-1$ spherical fibration) to spaces. We can promote the functor $T_n$ to a spectra valued functor by defining the *Thom spectrum* $T(\xi)$ to be

$$ T(\xi):= \Sigma^{-n}\Sigma^{\infty}T_n(\xi). $$

And again (by filtering the base space of the fibration by compact subspaces) $T$ can be viewed as functor from the slice category over the colimit space $BGL_1\mathbb{S}$ to the category of spectra.

There is another definition of Thom spectrum, under the hypothesis that the classifying map $B \to BGL_1\mathbb{S}$ is an infinite loop space map, which i found in these two papers AHR10 (def. 2.6), ABGHR14 (def. 4.2) and in the book "Formal Geometry and Bordism Operations" by E.Peterson Pet (Lemma A.4.1), which present the Thom spectrum in the following way. Write $\mathbb{S}$ for the sphere spectrum and take a map of connective spectra $f: b \to bgl_1\mathbb{S} = \Sigma gl_1\mathbb{S}$, where $gl_1$ is the right adjoint of the stabilization functor $\Sigma^\infty_+\Omega^\infty$. We can view $f$ also as a map of infinite loop space $$ \Omega^\infty f: \Omega^\infty b \to BGL_1\mathbb{S}$$ and hence as a stable spherical fibration over the base space $B=\Omega^\infty b$. Now the Thom spectrum of this spherical fibration is the homotopy pushout in the category of $E_\infty$ spectra of the following diagram $\require{AMScd}$ \begin{CD} \ \Sigma^\infty_+\Omega^\infty gl_1\mathbb{S} @>{\epsilon}>> \mathbb{S}\\ @V{\Sigma^\infty_+\Omega^\infty \lambda}VV @VVV\\ \Sigma^\infty_+\Omega^\infty Cj @>>> T(\Omega^\infty f), \end{CD}

where:

- $\epsilon$ is the counit map in the adjunction $\Sigma^\infty_+\Omega^\infty/ gl_1$
- $Cj$ is the cone of the desuspension $\Sigma^{-1}f$ and $\lambda$ is the map of the cofiber sequence $\require{AMScd}$ \begin{CD} \ \Sigma^{-1}b @>{j}>> gl_1\mathbb{S} @>{\lambda}>> Cj \end{CD}

**Question:**
Is there any motivation for the "homotopy pushout" definition of Thom spectrum to be equivalent to the classic definition for stable spherical fibration classified by map $B \to BGL_1\mathbb{S}$ that are infinity loop space map? I am facing big difficulties in reading the references that both of the papers and the book gives to motivate the latter definition of Thom spectra and i could not be able to find any other references.

**Further observations:** The only case in which the equivalence is clear to me is the case of the map $* \to bgl_1\mathbb{S}$ in which both definitions produce immediately the sphere spectrum $\mathbb{S}$. The general case is still obscure to me.

There should be another (equivalent?) definition (treated in ABGHR09) for the Thom spectrum of stable spherical fibration classified by map $f: BG \to BGL_1\mathbb{S}$, induced by group map $G \to GL_1\mathbb{S}$, in which $T(f)$ is presented as the "space of sections" of the sphere bundle over $BG$ associated to the map $f$. However, either this definition and how it should be related with the others is still not clear to me.