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Oct 2, 2012 at 5:00 comment added Benjamin Dickman 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
Oct 1, 2012 at 13:47 comment added Justin Young 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.
Oct 1, 2012 at 11:19 comment added Lennart Meier 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.
Oct 1, 2012 at 9:52 history asked Justin Young CC BY-SA 3.0