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

[Andrea Corradini <andrea.corradini@unipi.it>]

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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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