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

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

[David Yetter]

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Dear Colleagues:

This is surely something well-known, but it is also opaque to search-engine queries.  It is well-known (and I've both used the result and proved it by hand) than group objects in Cat and category object in Groups are the same thing:  strict monoidal categories in which every object and every arrow have an inverse with respect to \otimes.

What class of theories (e.g. finite product, left-exact, finitely axiomatizable equational,...) have the property that category objects in their category of models are the same as models of the theory in Cat?  Citations would be welcomed.

The question came up in work with an old student of mine, and rather than spending time proving the result we'd like for the particular theory at hand, I thought it best to see if it followed from something in the literature.  Alas, all sensible keyword combinations give pages of irrelevant search results, so asking the community seemed the best way to proceed.

Thanks in advance.

Best thoughts,
D.Y.


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

[Dusko Pavlovic]

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maybe i am missing something, but it sounds like a variation on the theme of "a sheaf of rings is a ring of sheaves", which got grothendieck from schemas to toposes. but categories are simpler than sheaves. if a category is viewed as a left-exact functor from, say, a finite limit sketch C into Set, then the question becomes: for which D are the D-preserving functors into the category of C-preserving functors equivalent to the category of C-preserving functors into D-preserving functors? ie which functors preserve the C-limits? is there a subtlety that i am missing? -- dusko

On Thu, Mar 21, 2024 at 11:28 AM David Yetter <dyetter@ksu.edu<mailto:dyetter@ksu.edu>> wrote:
Dear Colleagues:

This is surely something well-known, but it is also opaque to search-engine queries.  It is well-known (and I've both used the result and proved it by hand) than group objects in Cat and category object in Groups are the same thing:  strict monoidal categories in which every object and every arrow have an inverse with respect to \otimes.

What class of theories (e.g. finite product, left-exact, finitely axiomatizable equational,...) have the property that category objects in their category of models are the same as models of the theory in Cat?  Citations would be welcomed.

The question came up in work with an old student of mine, and rather than spending time proving the result we'd like for the particular theory at hand, I thought it best to see if it followed from something in the literature.  Alas, all sensible keyword combinations give pages of irrelevant search results, so asking the community seemed the best way to proceed.

Thanks in advance.

Best thoughts,
D.Y.


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

[Michael Barr, Prof.]

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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.

Michael
________________________________
From: Dusko Pavlovic <duskgoo@gmail.com>
Sent: Thursday, March 21, 2024 7:18 PM
To: David Yetter <dyetter@ksu.edu>
Cc: Categories mailing list <categories@mq.edu.au>
Subject: Re: T-algebras in CAT v. categories in T-alg

maybe i am missing something, but it sounds like a variation on the theme of "a sheaf of rings is a ring of sheaves", which got grothendieck from schemas to toposes. but categories are simpler than sheaves. if a category is viewed as a left-exact functor from, say, a finite limit sketch C into Set, then the question becomes: for which D are the D-preserving functors into the category of C-preserving functors equivalent to the category of C-preserving functors into D-preserving functors? ie which functors preserve the C-limits? is there a subtlety that i am missing? -- dusko

On Thu, Mar 21, 2024 at 11:28 AM David Yetter <dyetter@ksu.edu<mailto:dyetter@ksu.edu>> wrote:
Dear Colleagues:

This is surely something well-known, but it is also opaque to search-engine queries.  It is well-known (and I've both used the result and proved it by hand) than group objects in Cat and category object in Groups are the same thing:  strict monoidal categories in which every object and every arrow have an inverse with respect to \otimes.

What class of theories (e.g. finite product, left-exact, finitely axiomatizable equational,...) have the property that category objects in their category of models are the same as models of the theory in Cat?  Citations would be welcomed.

The question came up in work with an old student of mine, and rather than spending time proving the result we'd like for the particular theory at hand, I thought it best to see if it followed from something in the literature.  Alas, all sensible keyword combinations give pages of irrelevant search results, so asking the community seemed the best way to proceed.

Thanks in advance.

Best thoughts,
D.Y.


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

[John Baez]

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

[Dusko Pavlovic]

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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









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

[Ross Street]

But monoids for Cartesian product in abelian groups are still abelian groups too.

Remember which monoidal structure is involved in your monoid concept!

Ross

Sent from my iPhone

> On 23 Mar 2024, at 6:15 am, 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 groups.


----------

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

[Andrea Corradini]

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Dear David,

The following paper can be relevant:

J.W. Gray, The category of sketches as a model for algebraic semantics, Contemp. Math. 92 (1989)

Look at the tensor product in the category of sketches defined in Section 4.
Models of sketch $A \otimes B$ are models of $B$ in the category of models of $A$,
and $\otimes$ is commutative.

Best,
Andrea Corradini

On Thu, Mar 21, 2024 at 10:28 PM David Yetter <dyetter@ksu.edu<mailto:dyetter@ksu.edu>> wrote:
Dear Colleagues:

This is surely something well-known, but it is also opaque to search-engine queries.  It is well-known (and I've both used the result and proved it by hand) than group objects in Cat and category object in Groups are the same thing:  strict monoidal categories in which every object and every arrow have an inverse with respect to \otimes.

What class of theories (e.g. finite product, left-exact, finitely axiomatizable equational,...) have the property that category objects in their category of models are the same as models of the theory in Cat?  Citations would be welcomed.

The question came up in work with an old student of mine, and rather than spending time proving the result we'd like for the particular theory at hand, I thought it best to see if it followed from something in the literature.  Alas, all sensible keyword combinations give pages of irrelevant search results, so asking the community seemed the best way to proceed.

Thanks in advance.

Best thoughts,
D.Y.


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

[Uwe Egbert Wolter]

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There is even an earlier paper from J.W. Gray:

Categorical aspects of data type constructions in TCS 50  (1987) 103-135

Best

Uwe
________________________________
From: Andrea Corradini <andrea.corradini@unipi.it>
Sent: Friday, March 22, 2024 10:07
To: David Yetter <dyetter@ksu.edu>
Cc: Categories mailing list <categories@mq.edu.au>
Subject: Re: T-algebras in CAT v. categories in T-alg

Dear David,

The following paper can be relevant:

J.W. Gray, The category of sketches as a model for algebraic semantics, Contemp. Math. 92 (1989)

Look at the tensor product in the category of sketches defined in Section 4.
Models of sketch $A \otimes B$ are models of $B$ in the category of models of $A$,
and $\otimes$ is commutative.

Best,
Andrea Corradini

On Thu, Mar 21, 2024 at 10:28 PM David Yetter <dyetter@ksu.edu<mailto:dyetter@ksu.edu>> wrote:
Dear Colleagues:

This is surely something well-known, but it is also opaque to search-engine queries.  It is well-known (and I've both used the result and proved it by hand) than group objects in Cat and category object in Groups are the same thing:  strict monoidal categories in which every object and every arrow have an inverse with respect to \otimes.

What class of theories (e.g. finite product, left-exact, finitely axiomatizable equational,...) have the property that category objects in their category of models are the same as models of the theory in Cat?  Citations would be welcomed.

The question came up in work with an old student of mine, and rather than spending time proving the result we'd like for the particular theory at hand, I thought it best to see if it followed from something in the literature.  Alas, all sensible keyword combinations give pages of irrelevant search results, so asking the community seemed the best way to proceed.

Thanks in advance.

Best thoughts,
D.Y.


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

[Martti Karvonen]

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Dear David,

There is a paper by David B. Benson that characterizes exactly when two sketches S,T "commute" in the sense that S-models in T-mod are equivalent to T-models in S-mod: http://www.tac.mta.ca/tac/volumes/1997/n11/3-11abs.html<https://url.au.m.mimecastprotect.com/s/UxnSCP7L1NfYV3Y7izQUjO?domain=tac.mta.ca> . A convenient special case is that of limit sketches where this holds always (as limits commute with limits), so in particular any limit sketch will have this property wrt. categories.

Best,
Martti

On 22/03/2024 10:03, Uwe Egbert Wolter wrote:
There is even an earlier paper from J.W. Gray:

Categorical aspects of data type constructions in TCS 50  (1987) 103-135

Best

Uwe
________________________________
From: Andrea Corradini <andrea.corradini@unipi.it><mailto:andrea.corradini@unipi.it>
Sent: Friday, March 22, 2024 10:07
To: David Yetter <dyetter@ksu.edu><mailto:dyetter@ksu.edu>
Cc: Categories mailing list <categories@mq.edu.au><mailto:categories@mq.edu.au>
Subject: Re: T-algebras in CAT v. categories in T-alg

Dear David,

The following paper can be relevant:

J.W. Gray, The category of sketches as a model for algebraic semantics, Contemp. Math. 92 (1989)

Look at the tensor product in the category of sketches defined in Section 4.
Models of sketch $A \otimes B$ are models of $B$ in the category of models of $A$,
and $\otimes$ is commutative.

Best,
Andrea Corradini

On Thu, Mar 21, 2024 at 10:28 PM David Yetter <dyetter@ksu.edu<mailto:dyetter@ksu.edu>> wrote:
Dear Colleagues:

This is surely something well-known, but it is also opaque to search-engine queries.  It is well-known (and I've both used the result and proved it by hand) than group objects in Cat and category object in Groups are the same thing:  strict monoidal categories in which every object and every arrow have an inverse with respect to \otimes.

What class of theories (e.g. finite product, left-exact, finitely axiomatizable equational,...) have the property that category objects in their category of models are the same as models of the theory in Cat?  Citations would be welcomed.

The question came up in work with an old student of mine, and rather than spending time proving the result we'd like for the particular theory at hand, I thought it best to see if it followed from something in the literature.  Alas, all sensible keyword combinations give pages of irrelevant search results, so asking the community seemed the best way to proceed.

Thanks in advance.

Best thoughts,
D.Y.


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