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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asked Apr 2, 2012 at 3:11
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$\begingroup$ @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 MathewCommented Apr 3, 2012 at 0:30
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$\begingroup$ 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 MathewCommented Apr 3, 2012 at 0:36
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$\begingroup$ @David: I guess I mean "algebra over an $E_\infty$-operad" -- I definitely want more than something in the homotopy category (i.e. a homotopy commutative H space). Yes, it does seem that constructing model structures on commutative monoid objects is quite difficult. For instance, Tyler Lawson gave a nice argument that this can't be done for cdgas in characteristic $p$: see mathoverflow.net/questions/23269/… $\endgroup$– Akhil MathewCommented Apr 4, 2012 at 2:20
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$\begingroup$ @David: Glad to have been helpful! I'd be curious to eventually read it and learn more about this stuff... $\endgroup$– Akhil MathewCommented Apr 7, 2012 at 3:02
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2 Answers
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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 Barrat-Eccles operad. $\endgroup$ Commented Jun 29, 2018 at 8:46