in the diagram , equivalently, to the space of trivializations $$f^{*}\mathscr{L} \to R_{X}.$$

This definition also implies the following characterization, which plays a role in §\[sec:comp-thom-spectra\] when we compare our approaches to Thom spectra. Recall that $\operatorname{Aut}({R^{\circ}}) = {{{R}\text{-}\mathrm{line}}}({R^{\circ}},{R^{\circ}})$ is a group in the ${\infty}$-categorical sense: it is a group-like monoidal ${\infty}$-groupoid, or equivalently a group-like $A_{{\infty}}$ monoid. In general if $G$ is a group-like monoidal ${\infty}$-groupoid, then it has a classifying ${\infty}$-groupoid $BG.$

\[t-thom-sp-is-R-mod-G-intro\] Let $G$ be a group-like monoidal ${\infty}$-groupoid. Then the Thom spectrum of a map $BG {\xrightarrow{}} {{{R}\text{-}\mathrm{mod}}}$ is equivalent to the homotopy quotient ${R^{\circ}}/ G$.

The Proposition follows immediately from the construction of the Thom spectrum, since by definition the quotient in the statement is the colimit of the map of ${\infty}$-categories $$BG {\xrightarrow{}} {{{R}\text{-}\mathrm{line}}}\to {{{R}\text{-}\mathrm{mod}}}.$$ In these introductory remarks, we have taken some care to relate the ${\infty}$-categorical treatment to other approaches. In fact, as explained in [@math.CT/0608040; @DAGI; @DAGII], it is possible to develop the ${\infty}$-categories of of spectra, algebras in spectra, and $R$-modules entirely in the setting of quasicategories, without appeal to external models of spectra or even of spaces. We take this approach to Thom spectra and orientations in §\[sec:units-via-infty\]. It 