influence of groupoids on the category definition

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

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 "Einf??hrung 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









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