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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?

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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.

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  • $\begingroup$ This is perfect, thank you! $\endgroup$
    – Théo
    Aug 9 '21 at 21:53

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