From: Guillermo Cortiņas <gcorti@agt.uva.es>
Date: Feb 9, 2007 2:06 PM
Subject: Re: question about H-spaces
To: jim stasheff <jds@math.upenn.edu>
Cc: Don Davis <dmd1@lehigh.edu>, Brayton Gray <brayton@math.uic.edu>

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 <jds@math.upenn.edu> 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" <gcorti@agt.uva.es>
> > 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
> >
> >
>