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charset="utf-8" Content-Transfer-Encoding: quoted-printable On Fri, Mar 22, 2024 at 12:15=E2=80=AFPM Michael Barr, Prof. > wrote: I think Dusko is right. Monoids in the category of abelian groups are ring= s 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 amb= iguous. You can define monoids in any monoidal category, but what you get = depends on the monoidal structure. Monoids in AbGp with its cartesian pro= duct 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 fi= nite products will. If we define abelian groups and monoids this way, mo= noids in the category of abelian groups are the same as abelian groups in t= he 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 categor= y 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-categ= ory Lex that I mentioned last time. Best, jb You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --000000000000e02975061447039e Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable


On Fri, Mar 22, 2024 at 12:15=E2=80= =AFPM Michael Barr, Prof. <bar= r.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 g= roups.

The problem here is that "monoids in the category of abelian grou= ps" is ambiguous.  You can define monoids in any monoidal categor= y, 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 w= hich both abelian groups and monoids can be defined, and use that.  Th= e doctrine of monoidal categories won't work - but the doctrine of categori= es with finite products will.    If we define abelian groups and monoids this way, monoids in the category of abe= lian 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 pro= ducts, "models of A in the category of models of B in C" are equi= valent 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 Le= x that I mentioned last time.

Best,
jb




 
 
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