Date: Sat, 20 Dec 1997 09:53:31 -0400 (AST) Subject: non-Abelian categories Date: Fri, 19 Dec 1997 13:43:23 -0500 (EST) From: Colin McLarty Are there known axioms that stand to all groups the way the Abelian category axioms stand to Abelian groups? Date: Sun, 21 Dec 1997 16:08:25 -0400 (AST) Subject: Re: non-Abelian categories Date: Sat, 20 Dec 1997 14:21:35 GMT From: Michael Barr Colin's question, which essentially asks for a solution to the proportion abelian groups:groups :: abelian category:x does not of course have a unique answer. One solution was exact category and that was definitely one of the things I had in mind. In fact, I think I even said so. From my current vantage, I would add the following two properties: pointed and Mal'cev. For an equational category, that is almost enough to force a group structure (associativity is missing). I don't know how to force associativity by categorical properties, but pointed, exact and Mal'cev has to come awfully close to answering the question. Michael Date: Sun, 21 Dec 1997 16:09:42 -0400 (AST) Subject: Re: non-Abelian categories 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 Date: Sun, 21 Dec 1997 16:10:46 -0400 (AST) Subject: correction on category of groups Date: Sat, 20 Dec 1997 20:09:34 -0500 (EST) From: Colin Mclarty In an earlier post today I misdescribed a way of axiomatizing the category of groups by the triple for groups over sets. The point is that you can axiomatize the category of sets and the Eilenberg-Moore category for the triple for groups over it, and then identify the category of sets with the non-full subcategory of free groups and homomorphisms taking generators to generators; so that in a very narrow sense you would "only be talking about groups and homomorphisms". But really this amounts to defining groups as structured sets. What I want to know is, are there known axioms approaching the category of groups directly. Date: Mon, 22 Dec 1997 10:50:23 -0400 (AST) Subject: non-Abelian groups Date: Mon, 22 Dec 1997 10:33:36 +0100 (MET) From: Anders Kock Concerning Colin McLarty's questions: The category of groups should really be seen as a 2-category, in fact as part of the 2-category of groupoids. Date: Tue, 23 Dec 1997 10:24:27 -0400 (AST) Subject: Re: non-Abelian groups Date: Mon, 22 Dec 1997 11:34:35 -0500 (EST) From: Colin McLarty >Date: Mon, 22 Dec 1997 10:33:36 +0100 (MET) >From: Anders Kock > >Concerning Colin McLarty's questions: The category of groups should really >be seen as a 2-category, in fact as part of the 2-category of groupoids. > This gives an approach I had not thought of, and should have: take axioms for the category of categories, and change the description of the generator 2 to make it a groupoid. For the category of categories, 2 has no non-constant, non-identity endofunctors. For the category of groupoids it has exactly one, and that is an involution. (By a constant functor I mean one factoring through the terminal category 1.) A few other changes might be needed, depending on details of the axioms for the category of categories. But the key seems to be that the category of groupoids is cartesian closed and its insertion into the category of categories preserves exponentials--the prominent fact that a natural transformation with all components iso is a natural iso. So the center of the strength of the cat-of-cats axioms can be kept unchanged for the cat of groupoids. This is somehow orthogonal to the approach Mike Barr suggested, in that cartesian closedness of the category of categories is very close to associativity of composition. Mike lost associativity, but kept groups as opposed to groupoids. Anders's strategy gets associativity/cc-ness by foregoing uniqueness of objects. I have no idea how far this shows something objective, and how far it is an accident of the approaches we've thought of.