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
18-Jan-2002 08:31:43 -0400,2949;000000000001-00000000