From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/571 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: non-Abelian categories Date: Sun, 21 Dec 1997 16:08:53 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017047 26315 80.91.229.2 (29 Apr 2009 14:57:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:57:27 +0000 (UTC) To: categories Original-X-From: cat-dist Sun Dec 21 16:08:55 1997 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id QAA14618; Sun, 21 Dec 1997 16:08:53 -0400 (AST) Original-Lines: 62 Xref: news.gmane.org gmane.science.mathematics.categories:571 Archived-At: Date: Sat, 20 Dec 1997 13:41:28 -0500 (EST) From: Colin Mclarty Paul Taylor wrote to me Sat, 20 Dec > >> Are there known axioms that stand to all groups the way >> the Abelian category axioms stand to Abelian groups? > >This is a very cryptic question, Colin, why don't you say >in a bit more detail what you have in mind? I guess it was cryptic. And maybe it is trivial once spelled out. The thing is that I am writing a note on the Abelian category axioms as a foundation for the general theory of linear transformations, or of transformations linear over a given ring, etc. It is a reply to several of Sol Feferman's old complaints about categorical foundations which he recently affirmed unchanged on another e-mail list. In preparation I noticed that Emmy Noether used to look for "set theoretic foundations of group theory" by which she meant foundations that would NOT refer to elements or operations but would take the notion of quotient group as basic--and rely on her homomorphism and isomorphism theorems. By "group theory" she meant the study of varous categories. At least: the category of all groups, the category of groups with a fixed set of operators on them and homomorphisms prserving the operators, and the same for Abelian groups in place of all groups. Her Abelian groups with a fixed set of operators are in effect modules over a fixed ring. She was hugely attached to non-commutative algebras and to the generality of her proto-category-theoretic methods. She tends to present "all groups" and "commutative groups" as very similar things. I believe that most logicians today and philosophers of math also see these as quite similar, and that's who I'm writing for. So far as I see, they are not very similar categorically. The Abelian category axioms nicely suit Noether's goals for the Abelian cases--her homomorphism and isomorphism theorems become definitions and axioms on kernels and cokernels. I don't know anything comparable for all groups (or all groups with operators). You could axiomatize the category of all groups by, in effect, axioms for the category of sets (to be construed as free groups) plus the quotients given by the triple for groups over sets. And the same for groups with any set of operators. You could do the same for Abelian groups (or modules over fixed ring) but this is far less elegant than the Abelian category axioms with a projective generator--which you can then relate to set theory if you like by assuming completeness and that the generator is small. The triples approach axiomatizes completeness first, and the group structure as an add-on to it. Are there known axioms for the category of groups that do not in effect axiomatize the category of sets at the same time? Anything as elegant as the Abelian category axioms--though of course elegance is often in the eye of the beholder. Thanks, Colin