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Stream: learning: questions

Topic: ✔ enriched semicategories

Ralph Sarkis (Oct 04 2021 at 14:18):

I think I am interested in semicategories enriched in $([0,\infty],\geq)$ which would be non-symmetric diffuse metrics (the term is coined here), i.e.: $d:X\times X \rightarrow [0,\infty]$ satisfies only the triangular inequality. Hence, I was looking for references on enriched semicategories to see whether the category of $[0,\infty]$--semicategories and semifunctors (a category of diffuse metrics and non-expansive maps) has enough structure for my purposes (e.g.: products, coproducts, how monads work, etc.).

I saw some stuff here on regular semicategories and this follow-up paper but my parsing did not answer any of my questions.

Reid Barton (Oct 04 2021 at 14:40):

It looks like this category can be expressed as the category of models for a relational theory (based on the relations $R(x, y, a) := (d(x, y) \le a)$) or an essentially algebraic theory, so it is locally presentable.

Reid Barton (Oct 04 2021 at 14:51):

The main subtlety is that you need an axiom like "for any $x$, $y \in X$ and $a \ge 0$, if $R(x, y, b)$ for all $b > a$, then $R(x, y, a)$" which is not finitary.

James Deikun (Oct 04 2021 at 23:16):

You can also treat $[0,\infty]$ as a concrete category (of upper Dedekind cuts) and that makes it more amenable to standard tools ...

James Deikun (Oct 04 2021 at 23:34):

Also I would argue that the most natural notion of map for these things, from a (semi-)categorical perspective, would allow for points of zero self-distance to be mapped to Cauchy filters rather than particular points ... not sure what it would do for points of nonzero self-distance. (The maps I have in mind are basically adjunctions of enriched profunctors) ...

Ralph Sarkis (Oct 05 2021 at 12:52):

Thanks @Reid Barton , the term "relational theory" nudged me in the right direction, I found this paper which is precisely what I wanted to see.

Notification Bot (Oct 05 2021 at 12:58):

Ralph Sarkis has marked this topic as resolved.

Jules Hedges (Oct 05 2021 at 15:18):

Well that's a whole feature of Zulip I've never seen before

John Baez (Oct 05 2021 at 18:37):

This is the first time anything was resolved here!

Mike Shulman (Oct 05 2021 at 18:39):

It doesn't seem like a very Zulipy sort of thing to do. I associate "resolving" something with sites like stackoverflow and github. Here we mostly just talk, and lots of the time we go on talking about related (or unrelated) things after the original question has been answered.

Mike Shulman (Oct 05 2021 at 18:39):

(Of course, that was not intended as a criticism of Ralph for 'resolving' this thread!)

Fawzi Hreiki (Oct 05 2021 at 18:46):

I imagine it’s more intended for work or project related Zulips

John Baez (Oct 05 2021 at 18:48):

I've accidentally hit that button before, but I undid it.

Matteo Capucci (he/him) (Oct 06 2021 at 10:57):

if we ever have a thread on RH, we might use it and claim a million dollars