categories: Re: categorical incunabula

[Michael Barr <barr@barrs.org>]

I have never checked this out, but Fred Linton mentioned on more than one
occasion that Harald Bohr constructed his eponymous compactification of
abelian groups using a construction which was essentially the same as the
proof of the GAFT.  Take the product of "all" the compact groups generated
by a given group and then the closure of the subgroup there.  Actually, I
have always felt that the GAFT really doesn't tell you much that isn't
evident.  The SAFT, on the other hand, really does do something
non-trivial.

On Mon, 14 Jan 2002, Hans-E. Porst wrote:

> Concerning Peter Freyd's question
> 
> > Saunders's 1948 paper (without Sammy) first surprised me 40 years ago.
> > Buchsbaum in his 1955 paper that introduced abelian categories (under
> > the name "exact categories") said that he saw no way of defining
> > infinite products. Which meant that he hadn't seen Saunders's 1948
> > paper. Is this the first appearance of universal mapping definitions?
> 
> one certainly might consider A.A. Markov's definition of a free topological
> group (in 1945) as an earlier appearance:
> 
> A. A. Markov: On free topological groups, Izv. Akad. Nauk SSSR Ser. Mat. 9
> (1945), 3-64 [Amer. Math. Soc. Transl. 30 (1950), 11-88; Reprint: Amer.
> Math. Soc. Transl. Ser. I, 8 (1962), 195-272.]
> 
> Note, in this context, also the early apperances of what we now would call
> "applications (or predecessors) of Freyd's GAFT" (though none of these
> papers has the notions of category or functor)
> 
> S. Kakutani: Free topological groups and finite discrete product groups,
> Proc. Imp. Acad. Tokyo 20 (1944), 595-598
> 
> P. Samuel: On universal mappings and free topological groups, Bull. Amer.
> Math. Soc. 54 (1948), 591-598
> 
> In his 1957 paper
> 
> A. I. Malcev: Free topological algebras, Izv. Akad. Nauk SSSR Ser. Mat. 21
> (1957), 171-198 [Amer. Math. Soc. Transl. Ser. II, 17 (1961), 173-200.]
> 
> Malcev already begins his proof of the existence of a free topological
> algebra (as a topological subgroup of the corresponding product) with the
> phrase "In the usual way one can now prove".
> 
> -- 
> Hans-E. Porst                                 porst@math.uni-bremen.de
> FB 3: Mathematik                                Phone: +49 421 2182276
> University of Bremen                            Secr.: +49 421 2184971
> D-28334 Bremen                                  Fax:   +49 421 2184856
> 
> 
> 
> 
>