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

[Dusko Pavlovic <duskgoo@gmail.com>]

[-- Attachment #1: Type: text/plain, Size: 3677 bytes --]

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 question was about "categories in the category of algebras" and "algebras over categories". that places the questions in the realm of functorial semantics. functorial semantics has been developed in terms of product-preserving functors, finite-limit preserving 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 categories" do not involve tensor products and do not depend on the monoidal 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 carboni... and we used some stuff from joyal-street's tannakian categories. didn'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 stuff up :)))

all the best,
-- dusko

On Fri, Mar 22, 2024 at 12:05 PM John Baez <john.baez@ucr.edu<mailto:john.baez@ucr.edu>> wrote:


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









You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message.

View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Learn more about Microsoft 365 Groups<https://aka.ms/o365g>


[-- Attachment #2: Type: text/html, Size: 7971 bytes --]