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• CommentRowNumber1.
• CommentAuthorCorbin
• CommentTimeOct 7th 2021

Stub. Lots more work to do, but I don’t have the mental energy for it right now.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeOct 7th 2021

Lots more work to do, but I don’t have the mental energy for it right now.

You wrote a single sentence! :-)

And let’s get its grammar straight:

A meros (plural meroi) is a relational analogue of topoi.

Either of the following would work, grammatically (I don’t know contentwise, waiting to hear from you on that):

• The concept of a meros is a relational analogue of that of a topos.

• The concept of meroi is a relational analogue of that of topoi.

• A meros is much like a topos, but with Set, replaced by Rel.

Something like this.

• CommentRowNumber3.
• CommentAuthorCorbin
• CommentTimeOct 7th 2021

Explain I-categories. Still learning how to format stuff.

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeOct 7th 2021
• (edited Oct 7th 2021)

I suggest

A meros (plural meroi or meroses) is a relational analogue of a topos.

(assuming that the link to relation is appropriate here and you don't mean something else by ‘relational’). This has agreement of number between ‘meros’ and ‘topos’, anticipates that Greek plurals don't usually catch on in English (more people say ‘toposes’ than ‘topoi’ in my experience1), gives links, and saves boldface for the proper definition. While ‘the concept of’ is more logically precise (since it's the category of meroses and the category of toposes that will be analogous, not an individual meros and an individual topos), we can also save that level of precision for the definition too.

I was going to write this just now, but you were still editing.

1. And looking at the most famous example, not only do more people say ‘octopuses’ than ‘octopodes’, more people say the pseudo-Latin ‘octopi’ than ‘octopodes’, so I really think that Greek plurals in English are a lost cause. But feel free to use ‘meroi’ in the article itself.

• CommentRowNumber5.
• CommentAuthorCorbin
• CommentTimeOct 7th 2021

Define meroi.

• CommentRowNumber6.
• CommentAuthorCorbin
• CommentTimeOct 7th 2021

Sorry, I didn’t mean to take so long with my stub edit. I am not good with LaTeX. Also sorry for writing a single sentence and then falling asleep; I’ll avoid doing that in the future, as I recognize how rude it was.

I don’t understand meroi well enough to explain the precise connection between meroi and topoi; the connection might be vacuous beyond the relationship between Set and Rel. I don’t even know what kinds of functors connect meroi; are they geometric, logical, or something else? The bulk of the translation effort I’m doing here is in the usage of dagger-categories rather than Kawahara’s ♯ involution, and switching composition to right-to-left. I’m happy to embrace other nLab conventions too.

• CommentRowNumber7.
• CommentAuthorCorbin
• CommentTimeOct 7th 2021

Fix the first paragraph. I’m incorporating both Urs’ and Toby’s advice to fix the first sentence and also bulk up the analogy to topoi.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeOct 8th 2021
• (edited Oct 8th 2021)

In the lead-in paragraph I have fixed:

• “a meros is the analogue…” to “the concept of a meros is the analogue…”

• “morphisms of topoi” to “morphisms in topoi”

• “morphisms of meroi” to “morphisms in meroi”

Added mentioning of $Rel$ as the archetypical meros right in the lead-in paragraph, for that seems to give the quickest concrete idea for what is meant.

Hyperlinked many more keywords, notably relations, partial function, also the first appearances of topos.

Where it said “following Kawahara” I made it “following Kawahara 1995”, similarly further down when the name is mentioned in the Examples-section.

Moved the argument of universal quantifiers to \underset, in order to make it easier on the eye to discern what is being quantified over.

In the definition for “partial function” (here) I expanded

• “a morphism is a partial function…” to “a morphism in an I-category is called a partial function…”,

since otherwise it seems ambiguous. Hope that’s the intended reading, otherwise please fix.

Similarly, in the next line “A partial function is a total function” I added pointer back to the previous definition to make clear that “partial function” is meant in the sense just declared.

Similarly for the next definition of rational morphism, I expanded out to “a morphisms in an I-category is rational ” and after where it said “there exists functions” I added “in the sense of the previous definition”.

Similarly in the main definition, I have added pointers back to all the previous definitions (which seems important, since standard terms like “terminal object” and “quotient” seem to be used with non-standard meaning).

I have italicized all words in the sentences in which they are being defined.

Added missing punctuation in a number of places.

Is there more than Kawahara 95 on the topic? Has the suggestion been picked up yet by anyone?

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeOct 8th 2021

Not a great uptake.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 8th 2021
• (edited Oct 8th 2021)

How about sheaves with values in $Rel$, is that a “meros”?

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeOct 9th 2021

Is this just a variant axiomatization of “the bicategory of relations in a topos”, a.k.a. a power allegory?

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeOct 9th 2021

Or perhaps some other slightly different kind of allegory or bicategory of relations? Do we really need yet another name for it?

• CommentRowNumber13.
• CommentAuthorCorbin
• CommentTimeOct 12th 2021

Urs, I’m humbled by your editorial abilities. Thank you for your hard work. I’ll try to do better in the future.

There really should be some sort of analogy to category of presheaves. Offhand, I could imagine abusing the slogan that “a presheaf is a functor with an attitude”, and looking at a span of functors. From that latter page, “a relation is a correspondence which is (-1)-truncated”, or in other words, a relation is a (-1)-truncated span. So categories of spans should be meroi, given a nice enough 2-category.

David, Mike, I found Kawahara’s paper while trying to formalize Lojban, looking for ways to categorically talk about relations. The main question that I wanted to answer was the relationship between these three definitions of meros, allegory, and bicategory-of-relations. I wanted to write this page as a stopgap because I still don’t understand any of it well enough to decisively say that a meros is some specific flavor of allegory.

Overall, I am okay with this page meros existing or not existing. I am definitely paying attention to the implied notability standards for inclusion in nLab, given that I feel like it lands squarely within what to contribute; I will try not to contribute from non-notable papers.