For curiosity's sake, I have been reading a bit about the history of the development of spectra, and in particular modern categories of spectra such as EKMM S-modules and diagram spectra (e.g., symmetric and orthogonal spectra). As I understand it, one of the (many) motivations for the development of spectra was their role in computations arising from the Adams (and Adams-Novikov) spectral sequence. In the course of my reading, I've often stumbled across notation such as "$E^*E$" where $E$ is some ring spectrum such as $H\mathbb{Z}$ or $MU$. I'm having a bit of trouble parsing exactly what this notation means. So more generally:
Question: If $E$ and $X$ are spectra (in whichever category of spectra you like), what is generally meant by "$E^*(X)$"?
Thoughts: I assume this is essentially "the $E$-cohomology of $X$" but again I'm not entirely sure what that would be. If $X$ were a space, then I would think
\begin{equation}
E^*(X) = \oplus_n E_n^*(X) = \oplus_n[X, E_n]
\end{equation}
And if we consider homology rather than cohomology, and take $E$ and $X$ to both be spectra I would think
\begin{equation}
E_*(X) = \pi_*(E \wedge X)
\end{equation}
Is there some similar construction for $E^*(X)$ when $E$ and $X$ are both spectra? Perhaps involving a mapping spectrum, such as
\begin{equation}
E^*(X) = \pi_* \text{map}(X, E)
\end{equation}
when "map" is suitably defined and $E$ is appropriately fibrant?
Best Answer
Given two spectra $E$ and $X$, there is a function spectrum $F(X, E)$ such that $$\pi_* F(X, E) \cong E^{-*} X.$$ This spectrum represents the functor $Y \mapsto [X \wedge Y, E]$, so its existence, at least in the homotopy category, is guaranteed by the Brown representability theorem.
It is also possible to give constructions of function spectra in point-set models. For instance, in EKMM, this was done for "$\mathbb{L}$-spectra" in chapter I, which extends to $S$-modules (definition II.1.1). There is also an appendix in the book that contains more information about function spectra. For symmetric spectra, this is explained in example 3.38 in Schwede's book.
However, most of this is overkill if you just want to define $E$-cohomology. In this case, we can just define $E^n X = [X, \Sigma^n E]$ to be the group of homotopy classes of spectrum maps from $X$ to $E$. So, provided you know what maps and homotopies of spectra are, you can define cohomology. This is the approach taken in earlier accounts, such as Adams' blue book, which works with Boardman/sequential (CW) spectra.