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Antonin Delpeuch (Jan 20 2021 at 11:14):I am wondering why you want to call these things 2-rigs and not rig categories or bimonoidal categories? Especially given that 2-rig is already used for something else? When writing our paper on string diagrams for bimonoidal categories (so, your "2-rigs") it was difficult enough already to choose between "rig category" and "bimonoidal category", I hope we can avoid coming up with a new name in each paper on the subject!
fosco (Jan 20 2021 at 11:43):I agree it might be potentially confusing, and I should change it. "2-rig" is just shorter and easier to stick in my limited memory
Jules Hedges (Jan 20 2021 at 11:53):If I remember correctly we arbitrarily went with "bimonoidal category" mostly on the grounds that the word "rig" mildly irritates me. (And that was before a certain incident on twitter happened)
fosco (Jan 20 2021 at 11:58):Jules Hedges said:
the word "rig" mildly irritates me.
Oh, in that case I surely have to edit the entire manuscript!
fosco (Jan 20 2021 at 11:59):(related, but not https://www.urbandictionary.com/define.php?term=rigger )
Jules Hedges (Jan 20 2021 at 12:06):I'm torn between wanting to run with the joke, and my inner mod saying now is not the place
Jules Hedges (Jan 20 2021 at 12:06):(Link is mildly NSFW, but now I think about it, in corona time NSFW tags are probably totally unnecessary because nobody's at work)
fosco (Jan 20 2021 at 12:12):haha, I can remove it, np :grinning:
Jules Hedges (Jan 20 2021 at 12:28):Eh, I think it's ok
Nathanael Arkor (Jan 20 2021 at 13:15):Antonin Delpeuch said:
I am wondering why you want to call these things 2-rigs and not rig categories or bimonoidal categories? Especially given that 2-rig is already used for something else? When writing our paper on string diagrams for bimonoidal categories (so, your "2-rigs") it was difficult enough already to choose between "rig category" and "bimonoidal category", I hope we can avoid coming up with a new name in each paper on the subject!
2-rig (or 2-semiring, pseudorig, etc.) seems like the most consistent naming choice given existing categorical terminology. I find it frustrating when people use names that already have a well-defined meaning according to convention, and then co-opt it for their own use (2-rig, cartesian bicategory, costrength, etc.). It's hard enough to learn the terminology as it is. I don't think someone having already used a name in a manner inconsistent with convention is reason not to use a name for the meaning that matches convention, as long as this is clarified in the text.
Amar Hadzihasanovic (Jan 20 2021 at 13:27):I think “rig/semiring category” matches “monoidal category” much better?
I would match 2-rig or pseudorig with “pseudomonoid”, so that we would have something like
A rig category is a pseudorig internal to the monoidal bicategory of categories.
Amar Hadzihasanovic (Jan 20 2021 at 13:28):For the same reason I don't like “bimonoidal” because bimonoid is another name for a bialgebra, so I would expect it to be something like a pseudo-bialgebra in Cat.
Amar Hadzihasanovic (Jan 20 2021 at 13:30):Which I think is the same thing as a monoidal category because Cat is cartesian.
Nathanael Arkor (Jan 20 2021 at 13:31):Amar Hadzihasanovic said:
I think “rig/semiring category” matches “monoidal category” much better?
I would match 2-rig or pseudorig with “pseudomonoid”, so that we would have something like
A rig category is a pseudorig internal to the monoidal bicategory of categories.
I agree that I would expect the terminology to be used in the generality of a 2-category.
Nathanael Arkor (Jan 20 2021 at 13:32):Rig category isn't bad, but it would be nice if "rig" had an adjectival form (that is, we don't call them "monoid categories").
Jules Hedges (Jan 20 2021 at 13:35):"rigged"
Fawzi Hreiki (Jan 20 2021 at 13:38):I think rig isn’t too bad
Fawzi Hreiki (Jan 20 2021 at 13:38):It’s pretty intuitive: ring without negatives
Amar Hadzihasanovic (Jan 20 2021 at 13:38):Yes, and Ring = rig with negatives. It's recursive like GNU/Hurd.
Fawzi Hreiki (Jan 20 2021 at 13:39):But traditionally rig-category means Cartesian and cocartesian bimonoidal right?
Fawzi Hreiki (Jan 20 2021 at 13:39):Ie a category with products and sums
Fawzi Hreiki (Jan 20 2021 at 13:39):Usually distributive
Amar Hadzihasanovic (Jan 20 2021 at 13:41):According to the nLab rig category = bimonoidal category (no assumption of cartesian&cocartesian)...
Amar Hadzihasanovic (Jan 20 2021 at 13:41):(With distributivity of one over another)
Nathanael Arkor (Jan 20 2021 at 13:41):Fawzi Hreiki said:
But traditionally rig-category means Cartesian and cocartesian bimonoidal right?
That's typically called a bicartesian category (distributive category when they distribute).
Fawzi Hreiki (Jan 20 2021 at 13:42):The ‘bi’ prefix is so overloaded in category theory it’s kind of a tough one
Amar Hadzihasanovic (Jan 20 2021 at 13:42):Yeah, it's a mess.
Fawzi Hreiki (Jan 20 2021 at 13:43):Truthfully, I think bi- to mean weak 2- should just fade out
Fawzi Hreiki (Jan 20 2021 at 13:43):Since 2-.. usually means weak by assumption nowadays right?
Jules Hedges (Jan 20 2021 at 13:43):Is there an interesting relationship between bimonoidal categories and monoidal bicategories, so we can say that moving around the "bi-" prefix in the name actually corresponds to something
Fawzi Hreiki (Jan 20 2021 at 13:44):The philosophy of higher categories has kind of shifted to ‘as weak as possible unless otherwise specified’
Amar Hadzihasanovic (Jan 20 2021 at 13:44):I'm afraid bicategory vs 2-category as weak vs strict has stuck...
Amar Hadzihasanovic (Jan 20 2021 at 13:45):Personally I try to have “weak unless otherwise stated” for model-independent “notions”, but for specific models I try to just use whatever's been most commonly used.
Nathanael Arkor (Jan 20 2021 at 13:45):Obviously "bicategory" should mean a graph with two different category structures on it.
Fawzi Hreiki (Jan 20 2021 at 13:46):Hahahaha
Amar Hadzihasanovic (Jan 20 2021 at 13:46):Like “a higher category” is weak unless stated, but “2-category” is the strict algebraic model.
Fawzi Hreiki (Jan 20 2021 at 13:48):That’s a strange situation to be in though. I think the ‘2-‘ means strict convention primarily comes from the Australians since that’s what they tend(ed) to focus on
Fawzi Hreiki (Jan 20 2021 at 13:49):But for all other n, n-category usually just means weak as far as I’ve seen
Nathanael Arkor (Jan 20 2021 at 14:24):Nathanael Arkor said:
Obviously "bicategory" should mean a graph with two different category structures on it.
Incidentally, if you consider a "category enriched in a bimonoidal category" to mean "two enriched categories (one for each monoidal structure) with coincident objects and homs", then a graph with two category structures on it is exactly a category enriched in .
This doesn't seem pertinent to the original question though, because with this interpretation one cannot make sense of distributivity.
Joe Moeller (Jan 20 2021 at 17:12):Nathanael Arkor said:
Rig category isn't bad, but it would be nice if "rig" had an adjectival form (that is, we don't call them "monoid categories").
rigal sounds like wriggle
Nathanael Arkor (Jan 20 2021 at 17:19):In 50 years' time, when the etymology spawned here today has been forgotten, someone's going to open a MO question entitled: "What's so wriggly about distributive bimonoidal categories?"
Fawzi Hreiki (Jan 20 2021 at 18:19):Well obviously because string diagrams look like pasta
fosco (Jan 20 2021 at 21:05):Nathanael Arkor said:
In 50 years' time, when the etymology spawned here today has been forgotten, someone's going to open a MO question entitled: "What's so wriggly about distributive bimonoidal categories?"
Are we still having MO in 50 years? :grinning:
Antonin Delpeuch (Jan 22 2021 at 07:49):Nathanael Arkor said:
2-rig (or 2-semiring, pseudorig, etc.) seems like the most consistent naming choice given existing categorical terminology.
Do you know any papers which already use the term "2-rig" in this sense? I don't remember reading any.
Nathanael Arkor (Jan 22 2021 at 13:15):I haven't seen "2-rig" in many papers in any sense. I don't think anyone has studied these structures in full generality (i.e. not in Cat) yet.
John Baez (Jan 22 2021 at 15:58):In my paper on opetopes with James Dolan we used the term "2-rig" to mean a monoidal cocomplete category where the tensor product distributes over colimits - see the chart on page 16.
Nathanael Arkor (Jan 22 2021 at 16:05):I dislike this usage. The term "monoidally cocomplete category" is already in use to mean a cocomplete category with a tensor that distributes over colimits. I think, for X some algebraic structure, that "discrete 2-X in Cat" should be exactly an "X in Set".
Fawzi Hreiki (Jan 22 2021 at 17:06):Yeah that makes sense. A 2-rig is just a rig in .
Fawzi Hreiki (Jan 22 2021 at 17:08):(Up to coherence stuff)
Fawzi Hreiki (Jan 22 2021 at 17:14):Although then there’s the problem that a ‘2-category’ is not a category in .
fosco (Jan 23 2021 at 11:27):The thread veered into a discussion on what is the right name for these things :grinning:
I have questions, but I'll open another.
Morgan Rogers (he/him) (Jan 23 2021 at 11:32):If someone could come to some conclusion about terminology and explain it here and on the other topic, that would be great for future clarity.
John Baez (Jan 23 2021 at 17:08):Fawzi Hreiki said:
Yeah that makes sense. A 2-rig is just a rig in .
(Up to coherence stuff)
Historically, Kelly and Laplaza were the first to figure out the coherence stuff in this example, and they called a rig-up-to-coherent-natural-isomorphisms in CAT a ring category. This is unfortunate since they don't require the existence of a "subtraction" in their ring categories, which indeed is lacking in many interesting examples.
I agree that it would make more sense to call this, not a monoidally cocomplete category, a 2-rig.
John Baez (Jan 23 2021 at 17:09):The only problem with the term "monoidally cocomplete category" is that when you're writing a paper about them (as @Todd Trimble and @Joe Moeller and I now are), it gets really tiring to say "monoidally cocomplete category" once or twice in many sentences.
Jules Hedges (Jan 23 2021 at 17:21):An option I didn't see suggested yet is just that it's ok to have more than one name for something in parallel use. Particularly if it's simultaneously part of a larger pattern and also useful in isolation. It seems to me that calling it a "2-rig" if you're interested in higher algebra and n-things, and calling it a "rig category" if you merely happen to have 2 monoidal products lying around, isn't such a terrible thing
Jules Hedges (Jan 23 2021 at 17:30):Obviously it's better to not do that because it limits google-abiilty, but that can be mostly mitigated by flagging in big letters on nLab that there are multiple names in use
John Baez (Jan 23 2021 at 17:44):I think it's also okay to use a short name for convenience while admitting that one is leaving out some adjectives. For example, algebraic geometers almost always use "ring" to mean "commutative ring", because otherwise a lot of sentences would be packed with the word "commutative". Similarly, my paper with Todd and Joe uses "2-rig" to mean the kind of 2-rig we care about - a "symmetric monoidally cocomplete category" - just because it's shorter. I believe we include a suitable apology near the start.
John Baez (Jan 23 2021 at 17:44):In both cases one is using "X" to mean a specific kind of X that happens to be the kind that one is interested in, just for brevity.
Joe Moeller (May 06 2021 at 19:08):Nathanael Arkor said:
Morgan Rogers (he/him) said:
Joe Moeller said:
I wish there was an briefer way to distinguish different sorts of 2-rigs in a consistent and clear way.
A naming or notation system would be handy.
This was discussed extensively in the other thread. There are consistent names for each of these concepts, but "2-rig" is shorter, so people tend to overload it.
Right, you can be descriptive (eg Vect-enriched symmetric monoidally Cauchy complete), but it would be nice if there was a brief way to do it.
John Baez (May 06 2021 at 23:02):Just say vsmcc-2-rig. :upside_down:
John Baez (May 06 2021 at 23:06):I think similar stuff happens all the time in algebra, where they have semihereditary rings, Prufer rings, etc. In a talk or paper you'd usually start by saying something like
"Let be a Prufer ring."
or
"In this talk all schemes will be assumed locally noetherian."
John Baez (May 06 2021 at 23:07):So it's probably good to take a quite general concept of 2-rig as the official definition of "2-rig", and tack on extra adjectives to narrow it down.
John Baez (May 06 2021 at 23:09):@fosco mentioned something very general, which I believe Laplaza called a "ring category", which I call a rig category, and others call a "bimonoidal category".
John Baez (May 06 2021 at 23:09):There's no point in calling this a 2-rig since we already have a perfectly good name for it - actually up to 3 different names for it!
John Baez (May 06 2021 at 23:10):So it may be wise to save "2-rig" for the case where the additive monoidal structure is given by binary coproduct.
John Baez (May 06 2021 at 23:10):Then we can tack on other adjectives regarding whether it's enriched in some way, or has some larger class of colimits, etc.
Nathanael Arkor (May 06 2021 at 23:20):John Baez said:
fosco mentioned something very general, which I believe Laplaza called a "ring category", which I call a rig category, and others call a "bimonoidal category".
There's a more general definition, which is the most appropriate in my mind: namely a rig internal to a bicategory with two monoidal structures.
Nathanael Arkor (May 06 2021 at 23:20):A 2-rig in Cat would be what you call a rig category.
Notification Bot (May 06 2021 at 23:21):This topic was moved here from #theory: category theory > categorical differential equations by Nathanael Arkor
John Baez (May 06 2021 at 23:23):Okay. I think I've used "pseudorig" for the thing in a general bicategory.
Nathanael Arkor (May 06 2021 at 23:24):(I think it would be confusing to use "pseudorig" and "2-rig" to mean different things (apart from perhaps up to strictness), because usually "2-" and "pseudo-" mean the same thing up to strictness.)
Joe Moeller (May 06 2021 at 23:26):I think we used pseudorig to refer to a weakened rig-like object with respect to a cartesian bicategory, rather than two monoidal structures.
John Baez (May 06 2021 at 23:27):I need a short name for a kind of 2-rig where the additive monoidal structure comes from coproducts, because I rarely think about any other kind. So, left to my own devices, I'd call this a 2-rig.
One example not of this kind, which I often think about, is the groupoid of finite sets, with the usual + and , which however are not coproduct and product.
I conjectured that this is the (pseudo-)initial symmetric rig category, and that's apparently been proved.
David Michael Roberts (May 07 2021 at 01:03):Well, that proof's been published (here: https://doi.org/10.1016/j.jpaa.2021.106738), I will update the page!
David Michael Roberts (May 07 2021 at 01:08):Done!
John Baez (May 07 2021 at 04:03):Thanks!