In this setting, a map $X\to B{{GL_{1}}R}$ corresponds to a map of simplicial sets $$f: {{\Pi_{\infty}}X} \to {{{R}\text{-}\mathrm{line}}},$$ where ${{\Pi_{\infty}}X}$ is the ${\infty}$-groupoid (the singular complex) of the space $X$. By construction, ${{{R}\text{-}\mathrm{line}}}$ comes equipped with a universal bundle $\mathscr{L}$ of $R$-lines, and the map $f$ classifies the bundle of $R$-lines $f^{*}\mathscr{L}$ over $X.$ A lift in the diagram $$\label{eq:53}
\xymatrix{ & {{{{R}\text{-}\mathrm{triv}}}}
 \ar[d] \\
{{{\Pi_{\infty}}X}} \ar@{-->}[ur] \ar[r]_-{f} & {{{{R}\text{-}\mathrm{line}}};}  }$$ corresponds to an equivalence of bundles of $R$-lines $$f^{*}\mathscr{L} {\xrightarrow{{\simeq}}} R_{X},$$ where $R_{X}$ denotes the trivial bundle of $R$-lines over $X.$

In analogy to , we have the following.

\[def-7\] The Thom spectrum $Mf$ of $f$ is the colimit of the map of $\infty$-categories $${{\Pi_{\infty}}X} {\xrightarrow{f}} {{{R}\text{-}\mathrm{line}}}{\xrightarrow{}} {{{R}\text{-}\mathrm{mod}}}.$$

Note that the colimit is the same as the left Kan extension along the map to a point, so this definition is an analogue of the May-Sigurdsson description of the Thom spectrum as the composite of the pullback of a universal parametrized spectrum followed by the base change along the map to a point [@MR2271789 23.7.1,23.7.4]. Using this definition, it is straightforward to prove the following.

\[t-i-cat-obstr-intro\] The space of orientations of $Mf$ is weakly equivalent to the space of lifts 