### Summer Meanderings About Enriched Logic

#### Posted by David Corfield

Reading the recently appeared article

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

which treats Gabriel-Ulmer and related dualities in an enriched setting, I was wondering what sense we should make of “enriched logic”.

If, for instance, we may think of ordinary Gabriel-Ulmer duality as operating between essentially algebraic theories and their categories of models, then how to think of a finitely complete $\mathcal{V}$-category as a kind of enriched essentially algebraic theory?

That got me wondering about the case where $\mathcal{V}$ is the reals or the real interval, i.e., something along the lines of a Lawvere metric space, which led me to some recent work on continuous logic. This logic is associated with a longstanding program on continuous model theory, but it seems that the time is ripe now for category theoretic recasting, as in:

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

In this article Cho argues that the object of truth values of continuous logic is to be seen as a “continuous subobject classifier” in the sense of topos theory.

This raises questions of the proximity between real-enrichment and plain set-enrichment. Cho notes in his earlier thesis that the relevant thought is briefly discussed in the comments section of a post on the $n$-Category Café. This is the idea that $Set$ itself is a (subcategory of a) category of categories enriched in the poset of truth values, and came up in the discussion of Tom Avery’s post for the Kan Extension Seminar, starting with this comment.

Another sign that real-enriched category theory shares similarities to ordinary category theory is that we can have something close to conceptual completeness there.

- Jean-Martin Albert, Bradd Hart,
*Metric logical categories and conceptual completeness for first order continuous logic*, (arXiv:1607.03068)

Even though “Grothendieck’s notion of pre-topos is much too strong for the needs of continuous logic”, the authors arrive at the concept of a *metric* pre-topos.

The $n$-Café spilled over to a discussion on fuzzy logic as an enriched logic. I wonder if continuous logic has better categorical credentials.

And did Lawvere ever link his metric space ideas with his ideas on cohesion?

## Re: Summer meanderings about enriched logic

If we think of a category enriched over the monoidal poset $([0, \infty], \geq, +)$ as a Lawvere metric space, what does it mean for this enriched category to be finitely complete? I’m too lazy to work it out.