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.
Best Answer
The details of the comparison are treated in detail in the original ABGHR paper (and then unfortunately split in half across two papers in the updated version), so I'll just try to give a sketch of what's going on.
Thom spaces
Given any type of bundle $E \to B$ we can view this as a diagram of spaces indexed by $B$. The homotopy colimit of this diagram in the homotopy theory of spaces is just $E$ itself (which makes sense- we are 'summing up over $B$').
If $E \to B$ has a section $\infty: B \to E$, then this supplies each fiber with a basepoint. The homotopy colimit over $B$ of this diagram of pointed spaces can't be $E$, because it has a whole $B$-family of basepoints. So we need to identify those all together and take the cofiber $B \to E \to E/B$. The homotopy colimit is then $E/B$.
Example. Given a pointed spherical fibration, $E \to B$, with fibers $S^n$ (pointed!), this procedure gives the Thom space. So we learn that: the Thom space of a pointed spherical fibration is the homotopy colimit taken in the homotopy theory of pointed spaces of the family of spheres indexed over the base.
Example. Given an unpointed spherical fibration $E \to B$, with fibers $S^{n-1}$, we can form the fiberwise suspension by taking the cofiber $S^{n-1} \to CS^{n-1}$, fiberwise. This produces a pointed spherical fibration, and we can then take the Thom space as above. It's not hard to see that collapsing the section at $\infty$ of this suspended fibration gives the same answer as taking the mapping cone of the projection for the original fibration, whence the connection with your first definition.
Example. If $B = BG$ is the classifying space for a group, then the 'homotopy colimit indexed by $BG$' is also called the 'homotopy orbits for the action of $G$', since we may identify a bundle $E \to BG$ with a homotopy coherent action of $G$ on the fiber over a fixed basepoint of $BG$. So, in this case, the Thom space is of the form $S^n_{hG}$ where $G$ acts in some way on $S^n$. This includes the universal example, when $G = \mathrm{hAut}(S^n)$, the space of homotopy automorphisms of $S^n$ (i.e. the degree $\pm 1$ components of $\Omega^nS^n$ when $n>0$).
Thom spectra
For bundles of spectra it is nice to take the 'diagrams indexed by $B$' point of view as a definition. So a bundle is given by a map $B \to \{\text{space of spectra equivalent to }S^0\}=\mathrm{BGL}_1(S^0)$. This, in particular, defines a diagram of spectra and its homotopy colimit is the Thom spectrum. Since $\Sigma^{\infty}_+$ commutes with taking homotopy colimits (it's a left adjoint), we see that taking the suspension spectrum of a Thom space gives the Thom spectrum of the bundle obtained by taking fiberwise suspension spectra.
Now, in the case that $B = BG$, then we may identify functors $BG \to \mathsf{Sp}$ with modules over the $\mathbb{E}_1$-ring $S^0[G] := \Sigma^{\infty}_+G$. (This is a spectral version of the relationship between $G$-modules and $\mathbb{Z}[G]$-modules for an ordinary group). So we have:
spherical fibrations over $BG$ $\iff$ a coherent action of $G$ on the sphere spectrum $S^0$ $\iff$ an $S^0[G]$-module structure on $S^0$.
Under this correspondence, taking homotopy colimits corresponds to the construction on modules $M \mapsto M \otimes_{S^0[G]}S^0$ where we use the augmentation. (This is analogous to the statement that the coinvariants of a $G$-module are computed via a similar tensor product in the classical setting).
Thus, in the case $B = BG$, we may compute Thom spectra as $S^0 \otimes_{S^0[G]} S^0$ where the left $S^0$ has an interesting module structure, and the second $S^0$ is acted on through the augmentation.
Of course, the action of $G$ factors through the largest action of all of $\mathrm{GL}_1(S^0)$, so we can write this as
$S^0 \otimes_{S^0[G]}S^0 = S^0 \otimes_{S^0[\mathrm{GL}_1(S^0)]} S^0[\mathrm{GL}_1(S^0)] \otimes_{S^0[G]} S^0$.
In the special case when $G \to \mathrm{GL}_1(S^0)$ is 'normal', ie arises as the fiber of an $\mathbb{E}_1$-map $\mathrm{GL}_1(S^0) \to H$ (which happens int he infinite loop space context if we take $H = \Omega^{\infty}Cj$ in your notation), then we may simplify the right hand side of the tensor product as
$S^0 \otimes_{S^0[\mathrm{GL}_1(S^0)]} S^0[\mathrm{GL}_1(S^0)] \otimes_{S^0[G]} S^0=S^0 \otimes_{S^0[\mathrm{GL}_1(S^0)]} S^0[H]$.
So that's the relationship between the two definitions you write down.