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• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJul 9th 2019

which considers this and related dualities in the enriched setting.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeJul 9th 2019

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?

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJul 9th 2019
• (edited Jul 9th 2019)

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”.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJul 9th 2019

Re: #3: yes? (-:

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJul 9th 2019

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

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeAug 4th 2019

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

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