There are various strict monoidal model categories of spectra (e.g. symmetric spectra) where the honestly commutative monoid objects model the "coherently commutative" ring spectra (which might otherwise be expressed using, say, operads). Is there an analog for spaces? That is, there a monoidal model category, Quillen equivalent to spaces (in some monoidal sense), such that the category of commutative monoids in this category is (Quillen) equivalent to the category of algebras in spaces over some fixed and suitably free $E_\infty$operad? In spaces, this is false; topological abelian groups are very far from modelling infinite loop spaces.
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$\begingroup$ I'm a bit confused by this question. What do you mean by asking that the commutative monoid objects model the ''coherently commutative'' ring spectra? Also, by ''coherently commutative'' do you just mean $E_\infty$? Certainly strictly commutative monoids are $E_\infty$objects. Are you asking when these two concepts are exactly the same? If so, I believe the way to phrase this in terms of operads is to say that there is ''rectification'' between $Com$ and $E_\infty$. See for example the paper of Casacuberta, et al on Coloured operads $\endgroup$ – David White Apr 2 '12 at 20:01

$\begingroup$ Another question: you ask that the category of commutative monoids be (Quillen) equivalent to the category of algebras over some $E_\infty$ operad. In order to get a Quillen equivalence you'd need a model category structure. Why should the category of commutative monoids be a model category? It seems to me that this is a very restrictive hypothesis (and it's highly related to my current work), but maybe it's easy for Top. If so, I'd like to hear about it! $\endgroup$ – David White Apr 2 '12 at 20:01

$\begingroup$ @David: I mean "$E_\infty$" by "coherently commutative" (or any of the essentially equivalent concepts, e.g. commutative monoid objects in the $\infty$categorical sense. I'll have to look at the paper of Casacuberta. $\endgroup$ – Akhil Mathew Apr 3 '12 at 0:30

$\begingroup$ I think there are certain conditions when one gets a model structure on commutative monoid objects. It seems that there is a model structure on commutative monoid objects in certain cases, according to 4.3.2.1 in DAG III. I didn't really have anything too precise in mind when I referred to commutative monoids; all I meant was that there are plenty of infinite loop spaces which are not topological abelian groups or monoids. $\endgroup$ – Akhil Mathew Apr 3 '12 at 0:36

$\begingroup$ @Akhil: When you say $E_\infty$ do you just require that the diagrams commute up to homotopy and the coherence diagrams for those homotopies commute up to homotopy, etc? This may be different from what operad people mean by $E_\infty$ but I'm not sure. Thanks for the DAG reference. It doesn't apply here because Top is not combinatorial, but maybe something else does. The question of when $Monoids(M)$ is a model category is addressed in "Algebras and Modules in Monoidal Model Categories." It seems much harder to get a model structure on $CommMonoids(M)$, and this is something I do in my thesis. $\endgroup$ – David White Apr 3 '12 at 0:58
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Yes, such a model is developed in a paper of Blumberg, Cohen and Schlichtkrull about Thom spectra.
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$\begingroup$ This paper by Lind indeed compares different approaches, in particular $\ast$modules and $\mathcal{I}$spaces. The latter have been considered earlier by Schlichtkrull and in arxiv.org/pdf/1103.2764.pdf Sagave and Schlichtkrull give a more direct proof of the equivalence of commutative monoids in $\mathcal{I}$spaces with algebras over the BarratEccles operad. $\endgroup$ – Lennart Meier Jun 29 '18 at 8:46