From: "Ronnie Brown" <Ronnie@LL319dg.fsnet.co.uk>
To: "categories" <categories@mta.ca>
Subject: Re: groupoids versus homotopy 1-types
Date: Fri, 5 Jan 2007 17:06:06 -0000 [thread overview]
Message-ID: <5634.01768818057$1241019384@news.gmane.org> (raw)
I don't have a specific reference in 2-category language but the following
should be relevant:
The notion of a homotopy theory for groupoids was set up in my 1968 topology
book now revised and republished as Topology and Groupoids. In particular
\pi_1: spaces \to groupoids preserves homotopies. Fibrations were introduced
in an exercise, and developed in later editions. See also Philip Higgins'
Categories and Groupoids, (1971) now available as a TAC reprint. The nerve
and classifying space of a groupoid are in Graeme Segal's `Classifying
spaces and spectral sequences' (IHES) utilising Grothendieck's nerve of a
category. These preserve homotopy. The fact that for a CW-complex X, [X,BG]
\cong [\pi_1 X, G] is also well known: P.Olum Ann Math 1958?
There is also relevant material in Gabriel-Zisman's book, but I do not have
it with me.
People should also look at 2 papers on groupoids by P A Smith in the Annals,
1951.
Hope that helps.
Ronnie
www.bangor.ac.uk/r.brown
----- Original Message -----
From: "John Baez" <baez@math.ucr.edu>
To: "categories" <categories@mta.ca>
Sent: Wednesday, December 27, 2006 6:53 PM
Subject: categories: groupoids versus homotopy 1-types
> Dear Categorists -
>
> The following claim should be well-known (or false),
> but I don't know a reference:
>
> Let Gpd be the 2-category consisting of
>
> groupoids
> functors
> natural transformations
>
> and let 1Type be the 2-category consisting of
>
> homotopy 1-types
> continuous maps
> homotopy classes of homotopies
>
> where for present purposes "homotopy 1-types" means "CW complexes with
> vanishing higher homotopy groups regardless of the choice of basepoint".
>
> Claim: Gpd and 1Type are equivalent (or "biequivalent",
> in older terminology).
>
> In fact I bet there is an explicit pseudo-adjunction between them,
> with the "fundamental groupoid" 2-functor going one way and the
> "Eilenberg-Mac Lane space" 2-functor going the other way.
>
> Does anyone know for sure? Know a reference?
>
> Best,
> jb
>
>
>
>
>
>
>
> --
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>
next reply other threads:[~2007-01-05 17:06 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2007-01-05 17:06 Ronnie Brown [this message]
-- strict thread matches above, loose matches on Subject: below --
2006-12-27 18:53 John Baez
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