May and Sigurdsson provides one context for doing so, and they have discussed twisted generalized cohomology from this point of view in [@MR2271789]. The apparatus of $\infty$-categories provides another setting for such questions. In §\[sec:units-via-infty\] we show how to develop a framework for parametrized spectra in the context of the theory of quasicategories of Joyal and Lurie that makes it possible to develop the theory of Thom spectra and orientations essentially as we have described it above, with a construction of Thom spectra generalizing .

As mentioned above, it is remarkable the extent to which these ideas can and have been implemented using classical methods. To be more specific, the space $B{{GL_{1}}S}$ associated to the sphere spectrum is the classifying space for stable spherical fibrations, and if $A$ is an ${A_{\infty}}$ ring spectrum, then the unit $S\to A$ gives rise to a map $B{{GL_{1}}S} \to B{{GL_{1}}A}.$

Given a spherical fibration classified by a map $g: X\to B{{GL_{1}}S}$, we can form the solid diagram $$\label{eq:28}
\xymatrix{ {P} \ar[r] \ar[d] & {B (S,A)} \ar[r] \ar[d] & {E{{GL_{1}}A}}
 \ar[d]\\
{X} \ar[r]_-{g} \ar[ur] \ar@{<} `d [r] ` [rr]_-{f} [rr] & {B{{GL_{1}}S}}
\ar[r] & {B{{GL_{1}}A},} }$$ in which the rectangles are homotopy pull-backs. In that 