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Stream: deprecated: mathematics

Topic: Category of infinity-metric spaces (references?)

Franklin Pezzuti Dyer (May 08 2022 at 23:10):

I've been playing around with metric spaces lately, and it's occurred to me that infinity-metric spaces (i.e. metric spaces in which the distance function is valued in $\mathbb R^{\geq 0}\cup \{\infty\}$ rather than just $\mathbb R^{\geq 0}$) are potentially more interesting from a categorical perspective than typical metric spaces. For instance, if $\mathsf{Met}$ is the category of metric spaces with distance-non-increasing maps, and $\mathsf{InfMet}$ is the corresponding category of infinity-metric spaces, then $\mathsf{InfMet}$ has coproducts whereas $\mathsf{Met}$ does not.

Does anyone know if $\mathsf{InfMet}$ has been studied before? I'd be surprised if it hasn't - but I would appreciate it if someone could point me towards a book/paper/blog post that studies its properties. A cursory googling yields nothing.

Mike Shulman (May 08 2022 at 23:28):

Yes! Lawvere famously observed that extended (meaning infinite distances are allowed) quasi (meaning the metric need not be symmetric) pseudo (meaning distinct points can be at distance 0) metric spaces are exactly the same as categories enriched over the poset $[0,\infty]$, with the reverse of the usual ordering and addition as the tensor product. Hence, they have all the good properties that a category of enriched categories has, and moreover operations like Cauchy completion can be identified with categorical ones as well. The paper is Metric spaces, generalized logic and closed categories.

Mike Shulman (May 08 2022 at 23:28):

In his honor, extended quasi pseudo metric spaces are sometimes called [[Lawvere metric spaces]].

Franklin Pezzuti Dyer (May 08 2022 at 23:56):

This is incredible, I had no idea (and I'm not sure how I would have found out about this otherwise). Thanks so much!

Franklin Pezzuti Dyer (May 08 2022 at 23:57):

Although in retrospect, the nCatLab page on "metric space" seems like an obvious place to look :face_palm:

Ivan Di Liberti (May 09 2022 at 06:59):

1) Rosicky, Tholen. Approximate Injectivity
2) Adamek, Rosicky. Approximate injectivity and smallness in metric-enriched categories.
5) Rosicky. Metric Monads