# Lewis's convenience argument for $\mathbb{E}_{\infty}$-spaces

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:

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, $$\Gamma$$-spaces, $$\mathcal{I}$$-spaces) fare in such a list?

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.