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nLab > Latest Changes: metric space

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  1.  
    • 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.

    diff, v49, current

    • 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.

    diff, v50, current

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 6th 2019

    Thanks, Mike.

    • 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.

    diff, v52, current

    • 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 $[0, \infty]$".

  10. adding another definition due to Fred Richman

    Anonymous

    diff, v60, current

  11. adding section on metric spaces in homotopy type theory

    Anonymous

    diff, v61, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJan 31st 2024
    • (edited Jan 31st 2024)

    have hyperlinked the statement of the Urysohn metrization theorem here to its entry (Urysohn metrization theorem) and touched formatting and wording of the following (counter-)examples.

    diff, v68, current

    • CommentRowNumber13.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 28th 2025
    • (edited Jun 29th 2025)

    I saw this in the current version (compare #9):

    Imposing the symmetry axiom then gives us enriched †-categories. Note that when enriching over a cartesian monoidal poset, there is no difference between a †-category and a groupoid, so ultrametric spaces can also be regarded as enriched groupoids, which is perhaps a more familiar concept.

    (The requisite axioms for an enriched groupoid do not make sense when the enriching category is not cartesian, but one might argue that since in a poset “they would commute automatically anyway”, it makes sense to call any poset-enriched †-category also an “enriched groupoid”. However, perhaps it makes more sense just to speak about enriched †-categories.)

    I like the first sentence, because I know of other situations where one is enriching in a †\dagger-monoid and the condition hom(x,y) †=hom(y,x)\hom(x, y)^\dagger = \hom(y, x) turns out to be a really sensible one.

    But past the first sentence and before the last, I’m having trouble making sense of it (what “would commute automatically anyway”?), and I would like to have more eyeballs on this to help decide how much sense it makes or if we want to keep it in its current state. It’s not exactly written with a lot of confidence, after all. :-)

    • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeJun 29th 2025

    What about

    Imposing the symmetry axiom then gives us enriched †-categories. Note that when enriching over a cartesian monoidal poset, there is no difference between a †-category and a groupoid, so ultrametric spaces can also be regarded as enriched groupoids. But the requisite axioms for an enriched groupoid do not make sense when the enriching category is not cartesian, so an ordinary metric space cannot be regarded as an enriched groupoid.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 29th 2025

    That formulation would make much more sense to me.

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 30th 2025

    I went ahead and edited in a version of Mike’s suggestion.

    diff, v72, current

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