The existence of an $E_\infty$-diagonal is an obstruction for equipping a spectrum $E$  with the structure of a suspension spectrum. Conversely, in

Klein, J.R.: Moduli of suspension spectra. *Trans. Amer. Math. Soc.* **357**  (2005), 489–507

I showed that the existence of a suitably defined notion of $A_\infty$-diagonal on $E$ 
is equivalent to equipping $E$ with the structure of a suspension spectrum provided we are in the metastable range. Here "metastable" means $E$ is $r$-connected (for $r \ge 1$) and is weak
equivalent to a cell spectrum of dimension $\le 3r+2$. 

There are various elementary ways of defining the notion of $A_\infty$-diagonal, 
but they in the end
amount to the existence of a map $\delta: E \to (E\wedge E)^{\Bbb Z_2}$ (for a suitably
defined version of the smash product), which is a homotopy section to the map 
$(E\wedge E)^{\Bbb Z_2} \to E$ which is given by passing from categorical to geometric fixed points. The way I do this in the paper is the use the second stage of the Taylor tower
of the functor $E \mapsto \Sigma^\infty \Omega^\infty E$; this second stage turns out to be a model for $(E\wedge E)^{\Bbb Z_2}$.