# Reference request for K-Theory linearization

I posted this question on math.se here:https://math.stackexchange.com/q/2996787/482732, but I think it may be more appropriate here, sorry if I am wrong about that.

In Waldhausen's paper Algebraic K theory of Spaces(the long one) he proves the following:

$$A(X)\simeq \mathbb{Z}\times B\widehat{Gl}(\Omega^{\infty}\Sigma^\infty |G|)$$

Where $$|G|$$ is a loop group of $$X$$. My problem is that to apply $$B$$ we need $$\widehat{Gl}(\Omega^{\infty}\Sigma^\infty |G|)$$ to be $$A^\infty$$, but since $$\Omega^{\infty}\Sigma^\infty |G|$$ is only a ring up to homotopy this isn't obvious to me. Waldhausen just brushes past this point, so I was hoping to get a reference to somewhere that shows this in detail. Thanks.

• Uh, I might be a bit confused, but $\Omega^\infty\Sigma^\infty_+ |G|$ is certainly an $A_\infty$-ring – Denis Nardin Nov 13 at 15:49
• @DenisNardin I understand why both operations separately are $A^\infty$, I just don't understand the distributivity I suppose. – Noah Riggenbach Nov 13 at 15:57

I claim that for every $$A_\infty$$-space $$A$$, there is a canonical $$A_\infty$$-ring structure on $$\Omega^\infty\Sigma^\infty_+A$$.
First, $$\Sigma^\infty_+$$ from spaces to spectra is symmetric monoidal. So it sends an $$A_\infty$$-space $$A$$ to an $$A_\infty$$-algebra in spectra $$\Sigma^\infty_+A$$, that is an $$A_\infty$$-ring spectrum. The fact that $$\Sigma^\infty_+$$ is symmetric monoidal (at the model category/∞-category level) can be found in any modern book treating the smash product of spectra (e.g. it is proposition 4.7 in Elmendorf-May-Kriz-Mandell and corollary 4.8.2.19 in Lurie's Higher Algebra).
Secondly I claim that if $$E$$ is an $$A_\infty$$-ring spectrum, then $$\Omega^\infty E$$ has a canonical $$A_\infty$$-ring space structure. Exactly how this works will depend on your preferred definition of $$A_\infty$$-ring space, but it can be proven, e.g., with the same technique that May uses to prove the analogous statement for $$E_\infty$$-ring spaces (Corollary 7.5 in May's What precisely are $$E_\infty$$-ring spaces and $$E_\infty$$-ring spectra).
If all you care for is a construction of the $$A_\infty$$-structure on $$GL_1(\Sigma^\infty_+A)$$, I particularly like the approach in
where they identify $$GL_1(R)$$ with the automorphism group of $$R$$ as an $$R$$-module (and so it has an $$A_\infty$$-structure, since all automorphism groups in an ∞-category "trivially" do).