Re: T-algebras in CAT v. categories in T-alg

[John Baez <john.baez@ucr.edu>]

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On Fri, Mar 22, 2024 at 12:15 PM Michael Barr, Prof. <barr.michael@mcgill.ca<mailto:barr.michael@mcgill.ca>> wrote:
I think Dusko is right.  Monoids in the category of abelian groups are rings while abelian groups in the category of monoids are simply abelian groups.

The problem here is that "monoids in the category of abelian groups" is ambiguous.  You can define monoids in any monoidal category, but what you get depends on the monoidal structure.   Monoids in AbGp with its cartesian product are abelian groups, monoids in AbGp with its tensor product are rings.

To see commutativity of internalization, we should fix a doctrine in which both abelian groups and monoids can be defined, and use that.  The doctrine of monoidal categories won't work - but the doctrine of categories with finite products will.    If we define abelian groups and monoids this way, monoids in the category of abelian groups are the same as abelian groups in the category of monoids.  Both are simply abelian groups.

Indeed, for any categories A,B,C with finite products, "models of A in the category of models of B in C" are equivalent to "models of B in the category of models of A in C".  This is because the 2-category of categories with finite products is symmetric monoidal (pseudo)closed, just like the 2-category Lex that I mentioned last time.

Best,
jb









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