I am currently reading M. Mandel's paper `An Inverse $K$-theory Functor' to extract some results I may wish to use in my masters research project, and I am stuck on one particular claim.
On the top of the 3rd page Mandel claims that given a special $\Gamma$-space $X$, its associated infinite loop space is the group completion of $X(\mathbf{1})$.
Now, I know that given a special $\Gamma$-space $X$ the pointed simplicial set $X(\mathbf{1})$ can be equipped with a monoidal structure, up to homotopy. I also know what it means to (homotopy) group complete such a monoidal structure. Furthermore, I know that to $X$ we can associate a spectrum $X(\mathbb{S}^0),X(\mathbb{S}^1),X(\mathbb{S}^2),... $ (as constructed in Bousfield and Friedlander's Paper `Homotopy Theory of $\Gamma$-Spaces, Spectra, and Bisimplicial Sets'), which is stably equivalent to the $\Omega$-spectrum \begin{align} \Omega X(\mathbb{S}^1), \quad X(\mathbb{S}^1), \quad X(\mathbb{S}^2), \quad ... \end{align} The $0$-th space of this $\Omega$-spectrum is, of course, the $0$-th space of an infinite loop space.
My question is: Why is the space $\Omega X(\mathbb{S}^1)$ the group completion of $X(\mathbf{1})$?
I believe that $X(\mathbb{S}^0) \cong X(\mathbf{1})$, and we hence have the composite map \begin{align} X(\mathbf{1}) \to \mathbf{Hom}_*(\mathbb{S}^1,X(\mathbb{S}^1)) \to \mathbf{Hom}_*(\mathbb{S}^1,S |X(\mathbb{S}^1)| ) =: \Omega X(\mathbb{S}^1). \end{align} But I don't see any argument as to why the $X(\mathbf{1}) \to \Omega X(\mathbb{S}^1)$ is a group completion. What am I missing?
Thanks :)