Subject: non-Abelian groups

From: categories <[email-masked]>
Date: Mon, 22 Dec 1997 10:49:51 -0400 (AST)

Date: Mon, 22 Dec 1997 10:33:36 +0100 (MET)
From: Anders Kock <[email-masked]>

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.

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From: categories <[email-masked]>
Date: Tue, 23 Dec 1997 10:23:47 -0400 (AST)

Date: Mon, 22 Dec 1997 11:34:35 -0500 (EST)
From: Colin McLarty <[email-masked]>

>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.