The 1991 paper of Lewis, “*Is there a convenient category of spectra?*” proved that it is impossible to have a point-set model for spectra satisfying the following criteria:

- There is a symmetric monoidal smash product $\wedge$;
- We have an adjunction $\Sigma^\infty\dashv\Omega^\infty$;
- The sphere spectrum $\mathbb{S}$ is the unit for $\wedge$;
- There is either a natural transformation $$(\Omega^\infty E)\wedge(\Omega^\infty F)\Rightarrow\Omega^\infty(E\wedge F)$$ or a natural transformation $$\Sigma^\infty(E\wedge F)\Rightarrow(\Sigma^\infty E)\wedge(\Sigma^\infty F),$$ either of which is then required to satisfies the usual coherence conditions for monoidal functors.
- There is a natural weak equivalence $\Omega^\infty\Sigma^\infty X\dashrightarrow\lim_{n\in\mathbb{N}}(\Omega^n\Sigma^nX)$.

Since Lewis's paper, a number of model categories of spectra have appeared, each of which satisfies some, but not all, of the requirements (1)–(5). For instance, the category of $\mathbb{S}$-modules of Elmendorf–Kriz–Mandell–May satisfy (1)–(4), but not (5).

A modern point of view regarding spectra is that they are the $\infty$-categorical analogue of abelian groups in the sense that $\mathsf{Grp}_{\mathbb{E}_{\infty}}(\mathcal{S})\cong\mathsf{Sp}_{\geq0}$. Similarly, the $\infty$-categorical analogue of commutative monoids are $\mathbb{E}_{\infty}$-spaces, the $\mathbb{E}_{\infty}$-monoids in the symmetric monoidal $\infty$-category of spaces $\mathcal{S}$.

Is there an analogue of Lewis's argument for $\mathbb{E}_{\infty}$, giving a similar list of nice properties we may expect of a point-set model of $\mathbb{E}_{\infty}$-spaces, but such that there is no such point-set model satisfying all of them?

Moreover, in this case, how do the current point-set models for $\mathbb{E}_{\infty}$-spaces (such as $*$-modules, $\Gamma$-spaces, $\mathcal{I}$-spaces) fare in such a list?