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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeSep 6th 2011

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.

• CommentRowNumber2.
• CommentAuthorporton
• CommentTimeNov 23rd 2019

Added new research result that metric spaces are elements of a certain semigroup (or precategory) and that contractions are a special case of generalized continuous function.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeNov 23rd 2019

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).

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeNov 23rd 2019

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.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeDec 5th 2019

I added a remark about monoidal topology and bicategories of matrices to the section on Lawvere metric spaces, with a reference, and deleted the incomprehensible section.

• CommentRowNumber6.
• CommentAuthorDavidRoberts
• CommentTimeDec 6th 2019

Thanks, Mike.

1. Added subsection on category of metric spaces/short map.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJul 24th 2020

Skeletality of a Lawvere metric space is equivalent to the usual metric separation axiom only in the presence of symmetry.

• CommentRowNumber9.
• CommentAuthorDean
• CommentTimeApr 3rd 2021
I notice that the symmetry axiom makes a metric a groupoid enriched over $[0, \infty]$. That is, adding symmetry to a Lawvere metric turns "category enriched over $[0, \infty]$" into "groupoid enriched over $[0, \infty]$".