Two responses on group completions, from Peter May and Jim Stasheff....DMD _________________________________________________________- Date: Fri, 7 May 1999 13:25:58 -0500 (CDT) From: Peter May Subject: Still more on: Group completion: there are (at least) three things with the same name. 1. Classical: Grothendieck construction from monoids to groups. 2. Simplicial: Degreewise group completion from simplicial monoids to simplicial groups. 3. Homotopical: A map f: X >--> Y of H-spaces, say homotopy associative and commutative, is a group completion if the set of components of Y is a group, the map induced by f on sets of compenents is group completion in the classical sense, and the map induced by f in homology with coefficients in any commutative ring (equivalently, in any field) is localization at the multiplicative monoid of components of X (embedded in the zeroth homology group of X in the evident way) in the classical sense of commutative algebra. The May-Thomason uniqueness theorem for infinite loop space machines E requires a group completion in this last sense from the given structured space X to the zeroth space of the spectrum E(X): that is the only axiom that is required. Peter May ______________________________________________________ Date: Fri, 7 May 1999 14:15:52 -0400 (EDT) From: James Stasheff Subject: Re: more on group completion Beke writes: involves "group completion" in a sense that DIFFERS from that of Barratt at the time, it didn't seem so different especially if you interpret Moore's question as applying to topological monoids would like to know the status of moore's questions? (Recall Serre's distinction between a question and a conjecture.) .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds