Subject: fundamental groups
From: "Ronnie Brown"
Date: Thu, 13 Sep 2007 18:43:00 +0100
Kirsten,
Your construction is contained in the fundamental groupoid \pi_1 X but
there is no standard name except bundle of groups for the family of
vertex groups of a groupoid, or indeed for the family C(x,x) of monoids of
a category C. You could look in Mackenzie's book on Lie Groupoids.
To move from a groupoid to the family of vertex groups is throwing away
information, and the category of groupoids has many useful properties,
see
Topology and Groupoids
www.bangor.ac.uk/r.brown/topgpds.html
and Higgins' book
http://www.tac.mta.ca/tac/reprints/articles/7/tr7-2l.pdf
"I have known such perplexity myself a long time ago, namely in Van Kampen
type situations, whose only understandable formulation is in terms of
(amalgamated sums of) groupoids."
Alexander Grothendieck
Moving to fundamental groupoids also suggests the notion of higher
homotopy groupoids, which are hard to set up, but have powerful
applications, see the references in
www.bangor.ac.uk/r.brown/hdaweb2.htm
Ronnie Brown
> Subject: fundamental group family
> From: Kirsten Wickelgren
> Date: Wed, 12 Sep 2007 20:44:41 -0700
>
> Hello all,
>
> Ravi Vakil and I were wondering if anyone had an opinion about the name
of the following object: given a space X, pi_1(X,x) varies continuously
with x and one can glue the pi_1(X,x) together to form a group object over
X whose fibers are pi_1(X,x). Does this have a standard name? We've been
calling it by names like the fundamental group family, the fundamental
group bundle, the relative fundamental group, and the fundamental relative
group. Opinions?
>
> Thank you,
> Kirsten Wickelgren
>