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.

  • 1
    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
up vote 9 down vote accepted

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

Matthew Ando, Andrew J. Blumberg, David Gepner, Michael J. Hopkins, Charles Rezk An ∞-categorical approach to R-line bundles, R-module Thom spectra, and twisted R-homology

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).

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