case, $$Mf = (Mg){\wedge}A,$$ the space $B (S,A)$ is the space of $A$-oriented spherical fibrations, and to give an orientation $$Mg \to A$$ is to give a lift as indicated in the diagram.

In his 1970 MIT notes [@MR2162361] (in the version available at <http://www.maths.ed.ac.uk/~aar/books/gtop.pdf>, see the note on page 236), Sullivan introduced this picture, and suggested that Dold’s theory of homotopy functors [@MR0198464] could be used to construct the space $B (S,A)$ of $A$-oriented spherical fibrations. He also mentioned that the technology to construct the delooping $B{{GL_{1}}A}$ was on its way. Soon thereafter, May, Quinn, Ray, and Tornehave in [@MQRT:ersers] constructed the space $B{{GL_{1}}A}$ in the case that $A$ is an ${E_{\infty}}$ ring spectrum, and described the associated obstruction theory for orientations of spherical fibrations. Thus, at least for spherical fibrations and ${E_{\infty}}$ ring spectra, the obstruction theory for Thom spectra and orientations has been available for over thirty years.

Various aspects of the theory of units and Thom spectra have been revisited by a number of authors as the foundations of stable homotopy theory have advanced. For example, Schlichtkrull [@MR2057776] studied the units of a symmetric ring spectrum, and May and Sigurdsson [@MR2271789] have studied units and orientations 