Joyal’s theory of quasicategories, as developed in Lurie’s book [@math.CT/0608040], allows us to be precise about this. Specifically, we use this theory in §\[sec:units-via-infty\] and §\[sec:i-cat-ii\] to give an account of parametrized spectra (homotopy sheaves), Thom spectra and orientations which is very close to the intuitive picture discussed in §\[sec:prol-thom-isom\].

In the new theory, the analogue of $B{{GL_{1}}R}$ is ${{{R}\text{-}\mathrm{line}}}$, the subcategory of the $\infty$-category ${{{R}\text{-}\mathrm{mod}}}$ of $R$-modules consisting of equivalences of free rank-one cofibrant and fibrant $R$-modules. To see the virtues of ${{{R}\text{-}\mathrm{line}}}$, we note suggestively that it contains all two-simplices of the form $$\xymatrix{ {L} \ar[r]^{f} \ar[dr]_{h} & {M} \ar[d]^{g}
\ar@{}[dl]|(.32){\sigma}
\\
 & {N,} }$$ where $f,$ $g,$ and $h$ are $R$-module weak equivalences, and $\sigma$ is a homotopy of weak equivalences from $gf$ to $h.$ This is the sort of data mentioned at , and it exhibits ${{{R}\text{-}\mathrm{line}}}$ as the classifying space for “homotopy local systems of free $R$-modules of rank $1$” (we call them bundles of $R$-lines).

Specifically, for a fibrant simplicial set $X$, there is an equivalence of $\infty$-categories between maps $X \to {{{R}\text{-}\mathrm{mod}}}$ and bundles of $R$-modules over $X$. The identity map ${{{R}\text{-}\mathrm{mod}}}\to {{{R}\text{-}\mathrm{mod}}}$ classifies the “universal bundle of $R$-modules”, and pulling back along 