in that case we should require $A$ to be a cofibrant-fibrant algebra, and then (Proposition \[inf-t-pr-Rwe-and-GL\]) the homotopy type of ${{GL_{1}}A}$ is that of the subspace of the *derived* mapping space of $A$-module endomorphisms $\hom_{A} (A,A)$ consisting of weak equivalences. This approach is related to the approach studied in [@MR2271789].

For now we continue by imitating how one would proceed if ${{GL_{1}}A}$ were a topological group, acting on the spectrum $A.$ Associated to the group ${{GL_{1}}A}$ we would have the principal fibration $${{GL_{1}}A} \to E{{GL_{1}}A} \to B{{GL_{1}}A},$$ and given a map $f: X\to B{{GL_{1}}A}$, we would form the pull-back $$\begin{CD}
   P @>>> E{{GL_{1}}A} \\
   @VVV @VVV \\
    X @> f >> B{{GL_{1}}A}.
\end{CD}$$ One expects that the Thom spectrum associated to $f$ is a sort of Borel construction $$\label{eq:borelthom}
   Mf = P \times_{{{GL_{1}}A}} A,$$ and that the space of $A$-orientations of $Mf$ is the space of sections of $P/X$. To motivate this picture, consider the case that $G={{GL_{1}}R}$ is the group of units of a discrete ring $R$, so that $BG$, the moduli space of $G$-torsors, is equivalently the moduli space of free $R$-modules of rank $1$. To a principal $G$-bundle $$G \to P \to X$$ we can attach the 