# Simplicial model for $\Sigma^{\infty}_+ \Omega^{\infty}X$?

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

• Nick, why are you adding "+"? Isn't the relevant functor given by $X \mapsto \Sigma^\infty \Omega^\infty X$? – John Klein Jan 26 '18 at 13:50
• 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. – Nicholas Kuhn Jan 28 '18 at 18:29
• Oh yeah, I should have realized that....Well, so much for MO on the flu. – John Klein Jan 30 '18 at 1:40