3
$\begingroup$

It is a result of May's work on operads that the homotopy category (or $\infty$-category, if you prefer) of connective spectra is equivalent to a full subcategory of the category of representations of the little $\infty$-disk operad on the category $Top_+$ of pointed topological spaces (the full subcategory is then cut out by the condition of having inverse up to homotopy). Here the symmetric monoidal structure on $Top_+$ is given by smash product. This can be thought of as the homotopy-theoretic analogue of a definition of abelian groups as sets with some additive structure.

It is also of course true that the category of associative (or $E_n$) ring spectra is defined as representations of a suitable operad on the category of spectra, with symmetric monoidal structure the tensor product $\wedge$ operation. If we require the underlying spectrum to be connective, then (as far as I understand), we can work with this definition without leaving the connective-spectra category.

My question is whether there is some sort of generalization of an operad structure that gives ring spectra directly from $Top_+$, in a way similar to the definition of rings as sets with two operations.

The difficulty with this seems to be that the $\wedge$ operation is defined using some universal property (and, as far as I understand, doesn't admit a nice direct description on the level of $\Omega$ spectra): I'm curious if there are ways to get around this.

$\endgroup$
  • 5
    $\begingroup$ Chapter VII (and the beginning of Chapter VI) of May's LNM volume "$E_\infty$ Ring Spaces and $E_\infty$ Ring Spectra" is about doing exactly this: a recognition principle for connective $E_\infty$ ring spectra as spaces with two actions of the $E_\infty$ operad which distribute appropriately. $\endgroup$ – Eric Peterson Jul 6 '16 at 11:01
  • 4
    $\begingroup$ Thanks Eric. Let me advertise a modernized exposition without full details: math.uchicago.edu/~may/PAPERS/Final1.pdf. But one caveat: it is not ``the'' $E_{\infty}$ operad. It is crucial to use two suitably related $E_{\infty}$ operads, and I wish I knew more such operad pairs than I do. $\endgroup$ – Peter May Jul 6 '16 at 13:55
  • $\begingroup$ Thanks Peter! This looks very much like what I'm looking for (and yes, the question comes from thinking about equivariant homotopy theory) $\endgroup$ – Dmitry Vaintrob Jul 21 '16 at 14:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.