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Let $Sp^{\Sigma}$ be the category of symmetric spectra, and let $Sp^{\Sigma}-alg$ be the associated category of augmented commutative algebras. These are simplicial categories.

A question for experts: Can anyone give me a reference for, or explain a construction of, a simplicial functor $$ Sp^{\Sigma} \rightarrow Sp^{\Sigma}-alg$$ which looks like the functor $X \mapsto \Sigma^{\infty}_+ \Omega^{\infty}X$ in the homotopy category?

Nick Kuhn

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  • $\begingroup$ Nick, why are you adding "+"? Isn't the relevant functor given by $X \mapsto \Sigma^\infty \Omega^\infty X$? $\endgroup$
    – John Klein
    Jan 26, 2018 at 13:50
  • $\begingroup$ John: To have the suspension spectrum of an $E_{\infty}$ space, or more simply, a topological monoid, be a ring spectrum one needs to add a basepoint. (Think about it.) The output is then, in fact, a ring spectrum augmented over the sphere spectrum. By the way, I have answers to my question obtained in more traditional ways. $\endgroup$
    – Nicholas Kuhn
    Jan 28, 2018 at 18:29
  • $\begingroup$ Oh yeah, I should have realized that....Well, so much for MO on the flu. $\endgroup$
    – John Klein
    Jan 30, 2018 at 1:40

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