construction associates an $R$-module Thom spectrum to a map $B \to B{{GL_{1}}R},$ which need not arise from a spherical fibration. To compare to the classical situation, we suppose that $F$ arises from a stable spherical fibration via $$F: B {\xrightarrow{G}} B{{GL_{1}}S} {\xrightarrow{B{{GL_{1}}\iota}}} B{{GL_{1}}R}.$$ Then we can form the Thom spectrum $MG$ using Definition \[def-6\] (in §\[sec:comp-thom-spectra\] we show that this coincides with the Thom spectrum associated to $G$ as in for example [@LMS:esht]), and it follows directly from the definition that $$\label{eq:7}
    MF {\simeq}MG {\wedge}^{L} R.$$ We define an $R$-orientation of $MG$ to be a map of spectra $$MG \to R$$ such that the induced map of $R$-modules $$MF \to R$$ is an orientation as above. We then recover the result of Sullivan and of May, Quinn, Ray, and Tornehave [@MR2162361; @MQRT:ersers].

\[t-th-classical-orientation-sullivan-MQR\] Let $B (S,R)$ to be the pull-back in the solid diagram $$\xymatrix{ {P} \ar[r] \ar[d] & {B (S,R)} \ar[r] \ar[d] & {E{{GL_{1}}R}}
\ar[d]
\\
{B} \ar[r] \ar@{-->}[ur] & {B{{GL_{1}}S}} \ar[r] & {B{{GL_{1}}R}.}  }$$ Then the space of $R$-orientations of $MG$ is the space of indicated lifts.

Quasicategories and units {#sec:quasicatgories-units}
-------------------------

As we observed in §\[sec:prol-thom-isom\], the useful notion of ${{GL_{1}}R}$-bundle is that of a homotopy sheaf. 