The [1991 paper](https://www.sciencedirect.com/science/article/pii/0022404991900306#!) 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:


1. There is a symmetric monoidal smash product $\wedge$;
2. We have an adjunction $\Sigma^\infty\dashv\Omega^\infty$;
3. The sphere spectrum $\mathbb{S}$ is the unit for $\wedge$;
4. 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.
5. 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](https://arxiv.org/abs/0811.0553), [$\Gamma$-spaces](https://www.sciencedirect.com/science/article/pii/0040938374900226), [$\mathcal{I}$-spaces](https://arxiv.org/abs/1103.2764)) fare in such a list?



Lewis's convenience argument for $\mathbb{E}_{\infty}$-spaces
For $E\_\infty$ spaces, homotopy-theoretically there is a functor $L: \mathcal{S} \to E\_\infty \mathcal{S}$ with a right adjoint $R$. The only property on this list that really needs replacing on this list is property (5): the unit
$$
X \to RL(X)
$$
should be homotopy equivalent to the natural inclusion
$$
X \to Free\_{E\_\infty}(X) \simeq \coprod\_{k \geq 0} E \Sigma\_k \times\_{\Sigma\_k} (X^k)
$$
into the free $E\_\infty$-space on $X$. (Yes, yes, possibly a version with basepoints, I know)


I *believe* that all three of the models of $E\_\infty$ spaces that you list (commutative monoids in $\*$-modules, $\Gamma$-spaces, commutative $\mathcal{I}$-space monoids) satisfy properties (1)-(4) and fail the analogue of property (5), each due to an issue about whether an input to an adjunction is cofibrant/fibrant. For $\Gamma$-spaces, for example, the map $X \to RL(X)$ only adds a disjoint basepoint. Perhaps someone with more experience with the other models would be able to fill in those stories better.


