The idea that this should be so is due to the senior author (Hopkins), and was the starting point for this paper. For now we set $$\label{eq:30}
   Mf = {\mathbb{Z}}[P]\otimes_{{\mathbb{Z}}[G]} R,$$ and observe that with $T=R$ above we have $${(\mbox{$R$--modules})} (Mf,R) {\cong}{(\mbox{$G$--sets})} (P, R).$$ With respect to this isomorphism, the set of *orientations* of $Mf$ is to be the subset $$\xymatrix{ {{(\mbox{$R$--modules})} (Mf,R)} \ar@{-}[r]^-{{\cong}} &
{{(\mbox{$G$--sets})} (P, R)}
\\
{{(\mbox{orientations})} (Mf,R)} \ar@{-}[r]^-{{\cong}} \ar@{>->}[u] &
{{(\mbox{$G$--sets})} (P,G),} \ar@{>->}[u] }$$ which in turn is isomorphic to the set of sections of the principal $G$-bundle $P/X$, as expected.

Returning to the provisional definition of equation , one approach to generalizing to the input of a ring spectrum $A$ and a space $X$ is to develop the machinery to mimic that definition, in the form , in the setting of ${A_{\infty}}$ spaces. We carry this out in §\[sec:ainfty-thom-spectrum\].

Another possible approach to this problem involves coming to grips with homotopy sheaves of spectra, and construct $\xi$ as a homotopy sheaf of $A$-modules. If $X$ is a paracompact Hausdorff space, then as in [@MR0232393] or [\[HTT, 7.1\]]{}, it is equivalent to consider homotopy local systems of $A$-modules parametrized by $X.$

The parametrized homotopy theory of 