Gave an instance of the enriched version – meet semilattices and algebraic lattices.

]]>Admittedly it may not be a very “syntactic” kind of theory…

]]>Re: #3: yes? (-:

]]>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

- Simon Cho,
*Categorical semantics of metric spaces and continuous logic*, (arXiv:1901.09077)

with a “continuous subobject classifier”.

]]>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?

]]>I added the recent

- Stephen Lack, Giacomo Tendas,
*Enriched Regular Theories*, (arXiv:1907.02301)

which considers this and related dualities in the enriched setting.

]]>Fixed a dead link to the Lack-Power paper

Anonymous

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