groupoids and categories

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

Just to make a correction: I wrote "Brandy in 1926" instead of "Brandt
in 1926"!

To say more on this story, my survey article of 1987 writes:
-------------------------------------

Brandt???s de???nition of groupoid arose out of his work for over thirteen
years [6-10] on generalising to quaternary quadratic forms a composition
of binary quadratic forms due to Gauss [63].  Brandt then saw how to use
the notion of groupoid in generalising to the non-commutative case the
arithmetic of ideals in rings of algebraic integers, replacing the
classical ???nite abelian group by a ???nite groupoid [12].

--------------------------------------

[12]  is  H. BRANDT, ???Idealtheorie in Quaternionenalgebren???, Math. Ann.
99 (1928) 1-29.

A review by Baer of Jacobson's 1943 book on "The theory of rings" writes:

It  is shown  that  for  the  two-sided  ideals in  an  order  one may
obtain  unique  factorization  in  the  classical sense, including  the
commutativity  of  multiplication,  and  that under  comparatively
simple  necessary  conditions  on  the orders  in  the  ring  one
may  construct  Brandt's  groupoid  of  ideals.

------------------------

The 1939 Monograph by A A Albert, who was at Chicago,  may also say
something but I do not have it to hand.  Use  of groupoids in this area
was surely common knowledge in the 1940s.  I also came across published
correspondence of Hasse on this topic.

Ronnie Brown






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