From: Ronnie Brown <ronnie.profbrown-FhtRXb7CoQBt1OO0OYaSVA@public.gmane.org>
To: "categories-59hdLBrVOVU@public.gmane.org"
<categories-59hdLBrVOVU@public.gmane.org>,
algtop <algtop-l-wE+tr93vHrabo6XCN/16Dg@public.gmane.org>
Cc: John C Stillwell <stillwell-YPR7u1h/VASHXe+LvDLADg@public.gmane.org>
Subject: influence of groupoids on the category definition
Date: Sun, 15 Mar 2015 11:54:21 +0000 [thread overview]
Message-ID: <550572ED.9040303@btinternet.com> (raw)
A recent discussion on the category list has been continued on
http://mathoverflow.net/questions/199849/brandts-definition-of-groupoids-1926/199876
While trying to confirm my recollections of the interest of Reidemeister
in groupoids I did a web search on
Reidemeister 1932 Topologie, and to my pleasure saw on arxiv:1402.3906
Translation of Reidemeister's "Einführung in die kombinatorische
Topologie"
John Stillwell <http://arxiv.org/find/math/1/au:+Stillwell_J/0/1/0/all/0/1>
I am writing to advertise this translation. Reidemeister has a section
on "The groupoid", defines the fundamental groupoid, and also the action
groupoid corresponding to a group action.
I believe the next mention of groupoids in a topology text is by S-T Hu,
1964, which defines the fundamental groupoid, as does Spanier, 1966.
This led me in the early 1960s to think I ought to include something on
groupoids in the book I was writing. Then I came across Philip Higgins'
1964 paper on presentations of groupoids, which included a definition of
free products with amalgamation of groupoids; so I set an exercise on a
van Kampen type result. When I wrote out a solution of that, it seemed
so much better than my then current treatment that I decided to give a
full account. It still needed the notion of the fundamental groupoid on
a set of base points to get the appropriate general result, published in
1967. My text, now called "Topology and Groupoids", is still the only
topology text in English to give a van Kampen theorem in that setting.
See the discussion on
http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one/46808
In April 1967 G W Mackey introduced himself to me at a British
Mathematical Colloquium where I had given a talk on these results, and
he told me of his work on virtual groups and ergodic groupoids, which
involved the action groupoid of a group action. So I thought I ought to
do a chapter on covering spaces using the notion used by Higgins of
covering morphism of groupoids. Thus a covering map is modelled
algebraically by a covering morphism, which has advantages for results
on liftings of maps and morphisms. This fits of course with
Reidemeisater's action groupoid, which was used much later by Ehresmann
and Grothendieck.
The last Chapter of Reidemeister's book is on Branched Coverings. I
have often wondered if the use of groupoids can be helpful in that notion.
Ronnie Brown
_______________________________________________
ALGTOP-L mailing list
ALGTOP-L@lists.lehigh.edu
https://lists.lehigh.edu/mailman/listinfo/algtop-l
next reply other threads:[~2015-03-15 11:54 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2015-03-15 11:54 Ronnie Brown [this message]
2015-03-15 11:54 Ronnie Brown
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=550572ED.9040303@btinternet.com \
--to=ronnie.profbrown-fhtrxb7coqbt1oo0oyasva@public.gmane.org \
--cc=algtop-l-wE+tr93vHrabo6XCN/16Dg@public.gmane.org \
--cc=categories-59hdLBrVOVU@public.gmane.org \
--cc=stillwell-YPR7u1h/VASHXe+LvDLADg@public.gmane.org \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).