From: Guillermo Cortiņas Date: Feb 9, 2007 2:06 PM Subject: Re: question about H-spaces To: jim stasheff Cc: Don Davis , Brayton Gray Jim: thanks for your response. I see I must have formulated my question wrong, adding too many requirements. What I had in mind was: if H is a connected H-space then the shear map H\times H\to H\times H is a weak equivalence. If H is fibrant it is a strong equivalence, so H has an inverse. In the general case this applies to the fibrant replacement of H. Now assume H is a functorial connected H-space defined on some category I. If each H_i is fibrant, the argument above shows each H_i has an inverse, but maybe the inverse is not natural. This is not a problem if we assume H x H is both fibrant and cofibrant (in the closed simplicial model category of Bousfield-Kan). In the general case one would like to replace the given H by one which is both fibrant and cofibrant; the problem is that product of cofibrant objects need not be cofibrant. --Guillermo. On 2/9/07, jim stasheff wrote: > This is classic - if by H group you mean some strict group > structure or even some $A_\infty$-structure > > most natual counterexamnple is one of the exotic multiplications on S3 > the earliest example I think was a 2-stage Postinikov system > constructed for this purpose > > see my (mathematical) birth certificate > > jim > > > Don Davis wrote: > > Subject: Question on functorial H-spaces for Alg. Top. discussion list. > > From: "Guillermo Cortiņas" > > Date: Thu, 8 Feb 2007 19:32:51 +0100 > > > > Question: is it true that a functorial, $0$-connected, homotopy > > associative $H$-space with > > $H$-unit is naturally weakly equivalent to a functorial $H$-group? > > > > To formulate the question precisely: > > Let I be a small category; write $C$ for the category of functors from > > I to pointed simplicial sets. An $H$-associative $H$-unital funtorial > > $H$-space is an object $H$ of $C$ with an operation $+:H\times H\to > > H$ wich is associative up to natural homotopy > > and is such that $*$ is an identity up to natural homotopy. If in > > addition there is a natural > > map $H\to H$ which is naturally a homotopy inverse for the operation > > $+$, we call $H$ > > a functorial $H$-group. Assuming $H_i$ is $0$-connected for all $i\in > > I$, does it follow that there exists a functorial $H$-group $H'$ and a > > string of weak equivalences of $H$-objects connecting $H$ with $H'$ > > each of which preserves the operation up to natural homotopy? > > > > Guillermo Cortiņas. > > gcorti@agt.uva.es > > > > >