Does the group completion theorem apply to the James construction?

In other words, is the natural map $M \to \Omega B M$, for $M=JX$ the James construction on a space, a group completion? (By "group completion" I mean at the level of homology, I am aware of the space level version.) The versions of the group completion theorem that I have found such as Segal/McDuff have conditions on the monoid $M$, involving some kind of commutativity condition, and it is not clear to me that $JX$ satisfies any of these conditions. Could someone provide a reference or statement of a group completion theorem that clearly applies to $JX$, or is there actually a counterexample?

flag	asked Oct 1 at 9:52 Justin Young 163●3

If $X$ is connected, then $JX$ is connected, so there should be no problems. So, for a possible counter-example you could look at the case $X$ equal to the disjoint union of three points. – Lennart Meier Oct 1 at 11:19

I am particularly interested in the case when $X$ is not connected, otherwise there are many proofs out there. For discrete spaces, it seems to be true by direct calculation. – Justin Young Oct 1 at 13:47

Perhaps Section 6 of Classifying Spaces of Topological Monoids and Categories, Z. Fiedorowicz, American Journal of Mathematics, Vol. 106, No. 2 (Apr., 1984), pp. 301-350 would be useful to you. Stable URL: jstor.org/stable/2374307 – Benjamin Dickman Oct 2 at 5:00