Re: groupoids versus homotopy 1-types

Re: groupoids versus homotopy 1-types

[Ronnie Brown]

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
>
>
>
>
>
>
>
> -- 
> Internal Virus Database is out-of-date.
> Checked by AVG Free Edition.
> Version: 7.1.409 / Virus Database: 268.15.6/565 - Release Date: 02/12/2006
> 

groupoids versus homotopy 1-types

[John Baez]

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