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charset="utf-8" Content-Transfer-Encoding: quoted-printable hi john, i think mike used the example of "monoids in the category of abelian groups= " as an example of "algebras in the category of algebras". the original que= stion was about "categories in the category of algebras" and "algebras over= categories". that places the questions in the realm of functorial semantic= s. functorial semantics has been developed in terms of product-preserving f= unctors, finite-limit preserving functors, etc. caregories are finie-limit = preserving functors. there is no version, i think, of categories that are m= onoidal functors. so in the framework of the original question, there doesn't seem to be any = ambiguity. "categories of monoids" and "monoids over categories" do not inv= olve tensor products and do not depend on the monoidal structure. ((there is no such thing as "functorial semantics with respect to tensor pr= oduct". the correspondence between algebras for a monad and product-preserv= ing functors, referred to in the original question, does not lift to a corr= espondence with monoidal functors... pawel and i tried to develop a relatio= nal version of functorial semantics, and the only references that we could = find were two papers by aurelio carboni... and we used some stuff from joya= l-street's tannakian categories. didn't find much else and got stuck on bas= ic questions...)) i guess the facts that monoids as algebras for the monoid monad share the n= ame with algebras in a monoidal category is a terminological clash. a doubl= e terminological clash. so much for the hope of categories tidying stuff up= :))) all the best, -- dusko On Fri, Mar 22, 2024 at 12:05=E2=80=AFPM John Baez > wrote: 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 --0000000000001a530306145024a8 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable
hi john,

i think mike used the example of "monoids in the category of abel= ian groups" as an example of "algebras in the category of algebra= s". the original question was about "categories in the category o= f algebras" and "algebras over categories". that places the questions in the realm of functorial semantics. functorial semantics has b= een developed in terms of product-preserving functors, finite-limit preserv= ing functors, etc. caregories are finie-limit preserving functors. there is= no version, i think, of categories that are monoidal functors.

so in the framework of the original question, there doesn't seem to be= any ambiguity. "categories of monoids" and "monoids over ca= tegories" do not involve tensor products and do not depend on the mono= idal structure.

((there is no such thing as "functorial semantics with respect to= tensor product". the correspondence between algebras for a monad and = product-preserving functors, referred to in the original question, does not= lift to a correspondence with monoidal functors... pawel and i tried to develop a relational version of functorial semantics,= and the only references that we could find were two papers by aurelio carb= oni... and we used some stuff from joyal-street's tannakian categories. did= n't find much else and got stuck on basic questions...))

i guess the facts that monoids as algebras for the monoid monad share = the name with algebras in a monoidal category is a terminological clash. a = double terminological clash. so much for the hope of categories tidying stu= ff up :)))

all the best,
-- dusko

On Fri, Mar 22, 2024 at 12:05=E2=80= =AFPM John Baez <john.baez@ucr.edu<= /a>> wrote:


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