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I’ve removed this query box from metric space and incorporated its information into the text:
Mike: Perhaps it would be more accurate to say that the symmetry axiom gives us enriched $\dagger$-categories?
Toby: Yeah, that could work. I was thinking of arguing that it makes sense to enrich groupoids in any monoidal poset, cartesian or otherwise, since we can write down the operations and all equations are trivial in a poset. But maybe it makes more sense to call those enriched $\dagger$-categories.
As usual, I dislike the terminology, and the revision is very close to useless as it doesn’t explain the purported result; it just says to look at Porton’s book if you want to find out (shameless self-promotion I think is the usual phrase).
The terminology and notation are ugly and idiosyncratic (and the word “precategory” isn’t even defined), but if I understand correctly this looks like just a reinvention of the fact that metric spaces (like any enriched category) are monads (hence also particular morphisms) in a bicategory of profunctors. I think it would be reasonable to mention that fact, probably in the section on Lawvere metric spaces, though using standard terminology and notation of course. There are plenty of extant citations for generalizations of this point of view to other kinds of topological structures, including an entire book.
Thanks, Mike.
Added subsection on category of metric spaces/short map.
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