Subject: Re: two postings
Date: Fri, 17 Oct 2003 15:44:03 +0200
From: Rainer Vogt <rainer@mathematik.uni-osnabrueck.de>

> Subject: query
> Date: Thu, 02 Oct 2003 08:55:37 -0400
> From: jim stasheff <stasheff@email.unc.edu>
>
> Apparently Graeme's approach to infinite sloop spaces or rather
> to one fold loop spaces
> has be further abstracted to produce what is known as a `Segal category'
>
> at a quick glance, it seems to me these are related to Fukaya's A_\infty
>
> cats
> as my approach to \Omega X is related to Graeme's
>
> anyone seen this worked out or even commented on?
>
> jim

Sorry for the late reply: I have been on the road for two weeks.

In our work on the relation between homotopy
homomorphisms and the hammock localization of Dwyer and Kan,
published in Bol. Soc. Matematica Mexicana 37 (1992), Roland
Schwaenzl and I had to consider categories up to coherent
homotopies. We started off with A_\infty categories (there
are hand-written notes) but realized that we had less troubles
with Segal-categories, which we called \Delta -categories,
following Segal's notion of a \Delta -space. Both small Segal
categories and \Delta-categories can be rectified to strict
categories, so there is a direct comparison through the
strict version.
Take the inclusion of the category CAT of small topologically
enriched categories into the category of \Delta categories or
A_\infty categories, I am pretty sure that the analysis of the
rectification constructions implies that this inclusion is a
derived equivalence with respect to suitable model structures
on those categories.

Rainer Vogt