  The reduced universal monoid on the action category associated to a pointed
simplicial M-set has appeared in the guise of various simplicial monoid and
group constructions. These include the classical constructions of Milnor and
James, as well as their later generalizations by Carlsson and Wu. We prove
that, if any two n-simplices in the same orbit differ by the action of an
invertible monoid element, then the classifying space of this reduced universal
monoid is the homotopy cofiber of the inclusion from the pointed simplicial set
into its reduced Borel construction. The known formulae for the respective
classifying spaces of the above four constructions are special cases of this
result. Thus, we unify categorially the four above-mentioned constructions.
