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> 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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