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charset="utf-8"; format=flowed Content-Transfer-Encoding: quoted-printable Dear David, There is a paper by David B. Benson that characterizes exactly when two ske= tches 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 . A convenient special case is that of limit sketches where this hol= ds always (as limits commute with limits), so in particular any limit sketc= h 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 Sent: Friday, March 22, 2024 10:07 To: David Yetter Cc: Categories mailing list 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, Con= temp. 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=E2=80=AFPM David Yetter > 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 h= ave an inverse with respect to \otimes. What class of theories (e.g. finite product, left-exact, finitely axiomatiz= able equational,...) have the property that category objects in their categ= ory 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 s= pending 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 r= esults, so asking the community seemed the best way to proceed. Thanks in advance. Best thoughts, D.Y. 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 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 --------------fzS84t8t0GQVYuh0e07kU70b Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Dear David,

There is a paper by David B. Benson that characterizes exactly when two ske= tches S,T "commute" in the sense that S-models in T-mod are equiv= alent to T-models in S-mod: http://www.tac.mta.ca/tac/volumes/1997/n11/3-11abs.html . A convenient = special case is that of limit sketches where this holds always (as limits c= ommute with limits), so in particular any limit sketch will have this prope= rty 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 construct= ions 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, Con= temp. 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=E2=80= =AFPM David Yetter <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 \oti= mes.

What class of theories (e.g. finite product, left-exact, finitely axiomatiz= able equational,...) have the property that category objects in their categ= ory 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 s= pending 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 th= e best way to proceed.

Thanks in advance.

Best thoughts,
D.Y.
 
 
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
 
 
 
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   |   Leav= e group   |&n= bsp;  Learn more about Microsoft 365 Groups
 
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