the inclusion ${{{R}\text{-}\mathrm{line}}}\to {{{R}\text{-}\mathrm{mod}}}$ gives the the universal bundle of $R$-lines. We discuss this general approach to parametrized spectra in detail in Section \[sec:infinity-category-r\].

Now suppose that $R$ is a cofibrant and fibrant algebra in spectra. Let ${R^{\circ}}$ be an object of $R$-line, and let $$\operatorname{Aut}({R^{\circ}}) {\overset{\text{def}}{=}}{{{R}\text{-}\mathrm{line}}}({R^{\circ}},{R^{\circ}}) \subset {{{R}\text{-}\mathrm{mod}}}({R^{\circ}},{R^{\circ}})$$ be the subspace of weak equivalences. It is a group in the ${\infty}$-categorical sense: as the automorphisms of ${R^{\circ}}$ in the ${\infty}$-category ${{{R}\text{-}\mathrm{line}}}$, it is a group-like monoidal ${\infty}$-groupoid, or equivalently a group-like $A_{{\infty}}$ monoid. Note that ${{{R}\text{-}\mathrm{mod}}}({R^{\circ}},{R^{\circ}})$ is the derived space of endomorphisms of $R$, and so $\operatorname{Aut}({R^{\circ}})$ is the derived space of self weak equivalences of $R.$ We show (Proposition \[inf-t-pr-Rwe-and-GL\] and §\[sec:i-category-r\]) that $${{GL_{1}}({R^{\circ}})} {\simeq}\operatorname{Aut}({R^{\circ}}).$$

Now the full ${\infty}$-subcategory of ${{{R}\text{-}\mathrm{line}}}$ on the single object ${R^{\circ}}$ is just $B\operatorname{Aut}({R^{\circ}}).$ By definition, ${{{R}\text{-}\mathrm{line}}}$ is a connected ${\infty}$-groupoid (connected Kan complex), and so the inclusion $$B\operatorname{Aut}({R^{\circ}}) \to {{{R}\text{-}\mathrm{line}}}$$ is an equivalence, and it follows that $$B{{GL_{1}}({R^{\circ}})} {\simeq}{{{R}\text{-}\mathrm{line}}}.$$

The analogue of $E{{GL_{1}}R}$ is ${{{R}\text{-}\mathrm{triv}}}$, the $\infty$-category of trivialized $R$-lines: $R$-modules equipped with a specific equivalence to $R$. It is a contractible Kan complex, and the natural map $${{{R}\text{-}\mathrm{triv}}}\rightarrow {{{R}\text{-}\mathrm{line}}}$$ is a Kan fibration, our model for the fibration $E{{GL_{1}}R} \to B{{GL_{1}}R}.$

