Holmstrom Group completion

Ref Jardine-Goerss Thm 5.15 (p 255).

Thm: Suppose that $M \times X \to X$ is an action of a simplicial monoid $M$ on a simplicial set. Let $A$ be an abelian gp. Suppose further that the action of each vertex of $M$ induces an isomorphism in homology with A-coeffs. The the square of bisimplicial sets given by $X$, $EM \times_M X$, $pt$, $BM$ is homology cartesian. This means precisely that the map from $X$ to $hofib(d(\pi)))$ induces and iso in $A$-homology. Here $d$ denotes the diagonal and $\pi$ is the map $EM \times_M X \to BM$.

Applications: Analyze the output of infinite loop space machines. Example: Each connected component of the 0th space of the $\Omega$-spectrum corresponding to the sphere spectrum is a copy of the space given by the plus construction on the classifying space of the infinite symmetric group.

Another application: Describe the K-theory spectrum associated to a ring $R$.

http://mathoverflow.net/questions/36670/group-completions-and-infinite-loop-spaces

nLab page on Group completion