condition of their own, and so forth. Thus, we are brought quickly to the need for homotopical coherence machinery.

Units {#sec:units}
-----

The classical approach to addressing these issues goes as follows. Let $A$ be an associative ring spectrum, that is, a monoid in the homotopy category of spectra. Following [@MQRT:ersers], let ${{GL_{1}}A}$ be the pull-back in the diagram of (unpointed) spaces[^2] $$\label{eq:59}
\begin{CD}
{{GL_{1}}A} @>>> {\Omega^{\infty}}{A}
 \\
@VVV @VVV \\
(\pi_{0}A)^{\times} @>>> \pi_{0}A.
\end{CD}$$ If $X$ is a space, then $$[X,{{GL_{1}}A}] = \{f\in A^{0} ({X_{{+}}})| \pi_{0}f (X) \subset
(\pi_{0}A)^{\times}\} = A^{0} ({X_{{+}}})^{\times},$$ so ${{GL_{1}}A}$ is called the *space of units* of $A,$ and so a more refined definition of ${{\underline{A}}^{\times}}$ is $${{\underline{A}}^{\times}} (X) = \operatorname{map}(X,{{GL_{1}}A}).$$ If $A$ is ${A_{\infty}}$ ring spectrum in the sense of Lewis-May-Steinberger, then ${{GL_{1}}A}$ is a group-like ${A_{\infty}}$ space, so it has a delooping $B{{GL_{1}}A}$, and the appropriate cohomology group is $$H^{1} (X,{{\underline{A}}^{\times}}) = [X,B{{GL_{1}}A}].$$ Moreover if $A$ is an ${E_{\infty}}$ ring spectrum, then there is a spectrum ${{gl_{1}}A}$ such that $${{GL_{1}}A} {\simeq}{\Omega^{\infty}}{{gl_{1}}A}.$$

One could also work in a modern symmetric monoidal category of spectra, such as the $S$-modules of [@EKMM] or one of the categories of diagram spectra [@MR1806878]. As we discuss in §\[sec:space-units-derived\], 