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