Subject: non-Abelian categories

From: categories <[email-masked]>
Date: Sat, 20 Dec 1997 09:52:52 -0400 (AST)

Date: Fri, 19 Dec 1997 13:43:23 -0500 (EST)
From: Colin McLarty <[email-masked]>

        Are there known axioms that stand to all groups the way the Abelian
category axioms stand to Abelian groups?

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From: categories <[email-masked]>
Date: Sun, 21 Dec 1997 16:08:15 -0400 (AST)

Date: Sat, 20 Dec 1997 14:21:35 GMT
From: Michael Barr <[email-masked]>

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

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From: categories <[email-masked]>
Date: Sun, 21 Dec 1997 16:08:53 -0400 (AST)

Date: Sat, 20 Dec 1997 13:41:28 -0500 (EST)
From: Colin Mclarty <[email-masked]>

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