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I added the recent
which considers this and related dualities in the enriched setting.
If we may think of ordinary Gabriel-Ulmer duality as operating between essentially algebraic theories and their categories of models, how should we think of the enriched version between $\mathcal{V}-\mathbf{Lex}$, the 2-category of finitely complete $\mathcal{V}$-categories ($\mathcal{V}$-categories with finite weighted limits), finite limit preserving $\mathcal{V}$-functors, and $\mathcal{V}$-natural transformations, and $\mathcal{V}-\mathbf{Lfp}$, the 2-category of locally finitely presentable $\mathcal{V}$-categories, right adjoint $\mathcal{V}$-functors that preserve filtered colimits, and $\mathcal{V}$-natural transformations?
Can I think of a finitely complete $\mathcal{V}$-category as a kind of theory?
So in the case where $\mathcal{V}$ is the reals or the real interval, i.e., something along the lines of a Lawvere metric space, there appears to some connection to continuous logic
with a “continuous subobject classifier”.
Re: #3: yes? (-:
Admittedly it may not be a very “syntactic” kind of theory…
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