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