The article Justin refers to is ``Localization and completion theorems for $MU$-module spectra''. Annals of Math. 146(1997), 509-544. Its key technical tool is the multiplicative norm map. That is also a key technical tool in the work of Hill, Hopkins, and Ravenel on the Kervaire invariant problem. Their definition exhibits the norm map explicitly on the equivariant spectrum level, Greenlees and May only implicitly. Anna Marie Bohmann has lifted the Greenlees-May construction to the spectrum level and shown that the two coincide.

Thanks, Tom, of course that is another typo. It would be nice if comments could also be edited.

This is an edit of my comment above (since I don't know how to edit it properly): $\sma$ should be $\wedge$, $\rtarr$ should be $\to$.

In based spaces, $X\sma X$ is a quotient of $X\times X$, so we do have a diagonal, namely the composite of the diagonal $X\to X\times X$ and the quotient map $X\times X\to X\sma X$. Of course, it is not a categorical diagonal, but it is often useful. Since the suspension spectrum functor commutes with smash products, this does give suspension spectra a diagonal. It is used all the time in duality theory. A relevant conceptual point is that, in spectra, the canonical map $X\wedge X\rtarr X\times X$ is an equivalence, just as in Abelian categories.

Except that he perhaps exaggerates the utility of $\infty$-categories, I agree with Dylan's answer above. His point (1) applies to certain contexts, and the all'' in his (2) is way too strong. But his point is right: that is not what $\infty$ categories (or categories for that matter) are there for. One can of course specialize either to more calculable contexts (as in Ronnie's answer) but that misses the thrust of the question, I think. It would be nice if there were more young mathematicians who felt at home with both explicit computations'' and abstract theory.