Groups vs. groupoids (Pat Donaly)

Groups vs. groupoids (Pat Donaly)

[Jpdonaly]

To all category theorists:

While (partially) responding to Tom Leinster's 7/04 query regarding limit
preservation, a query of my own occurred to me: On pages 6 and 7 of Alain Connes'
book, "Noncommutative Geometry", he writes, "It is fashionable among
mathematicians to despise groupoids and to consider that only groups have an authentic
mathematical status, probably because of the pejorative suffix oid."
Professor Connes later cites the groupoid of states of the hydrogen atom in order to
eliminate the prejudice against groupoids, but, for group theorists, there is a
more direct way: Since Frobenius and/or Burnside adopted the concept of an
abstract group in order to consider general group actions and representations,
group theorists have been heavily involved in groupoids, whether they liked it
or not.

Is it generally understood by categorists that every group action---as a
comma category---is a groupoid?

Pat Donaly

Re: Groups vs. groupoids (Pat Donaly)

[Andree Ehresmann]

In answer to Pat Donaly

The connection between group actions and groupoids has been known and 
extensively used for a long time. It was realized by Charles Ehresmann in 
the early fifties. In fact Charles came to categories from groupoids, and 
to groupoids from group actions and from pseudogroups of transformations.

In particular, in his works on fibre bundles and Differential Geometry, he 
associated a groupoid to a pseudogroup of transformations (1), then 
considered action of groupoids of jets as extending group actions (2).

In the paper (3) he introduces topological and differentiable categories 
(i.e., internal to Top and to Diff), in view of associating to a principal 
bundle H a particular topological groupoid P (called a locally trivial 
groupoid). He then finds the locally trivial bundles associated to H as the 
spaces on which there is an (internal) action of this groupoid. Given a 
topological space F with an action of a sub-group of P, he constructs such 
a space with fibre F by an "enlargement"  process he had defined in his 
important paper (4).

These results and many others can be found in the series of papers 
reprinted in "Charles Ehresmann : Oeuvres completes et commentees" (more 
specially in Part I), 1980-83..

(1) Les prolongements d'une variété différentiable, Atti IV 
Cong.  dell'Unione Mate. Italiana, Taormina 1951, reprinted in "Oeuvres", 
Part I, pp. 207-215.
(2) Introduction à la théorie des structures infinitésmales et des 
pseudo-groupes de Lie, Actes Coll. Intern. Geom. Diff. Strasbourg, CNRS 
1953, reprinted in "Oeuvres", Part I, pp. 217-230.
(3) Categories topologiques et categories differentiables, Coll. Geom. 
Diff. Globale, CBRM Bruxelles 1959, reprinted in "Oeuvres", Part I, pp. 
237-250.
(4) Gattungen von lokalen Strukturen, Jahres. d. Deutsches Math. 60-2, 
1957, reprinted in "Oeuvres", Part II, pp. 125-153.

                         Andree C. Ehresmann