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September 15, 2020

Symmetric Pseudomonoids

Posted by John Baez

The category of cocommutative comonoid objects in a symmetric monoidal category is cartesian, with their tensor product serving as their product. This result seems to date back to here:

Dually, the category of commutative monoid objects in a symmetric monoidal category is cocartesian. This was proved in Fox’s suspiciously similar paper in Cocomm. Coalg.

I’m working on a paper with Todd Trimble and Joe Moeller, and right now we need something similar one level up — that is, for symmetric pseudomonoids. (For example, a symmetric pseudomonoid in Cat is a symmetric monoidal category.)

The 2-category of symmetric pseudomonoids in a symmetric monoidal 2-category should be cocartesian, with their tensor product serving as their coproduct. I imagine the coproduct universal property will hold only up to 2-iso.

Has someone proved this, so we don’t need to?

Hmm, I just noticed that this paper:

proves the result I want in the special case where the symmetric monoidal 2-category is Cat. Namely:

Theorem 2.3. The 2-category SMC of symmetric monoidal categories, strong monoidal functors, and monoidal natural transformations has 2-categorical biproducts.

Unfortunately their proof is not purely ‘formal’, so it doesn’t instantly generalize to other symmetric monoidal 2-categories. And surely the fact that the coproducts in SMC are biproducts must rely on the fact that Cat is a cartesian 2-category; this must fail for symmetric pseudomonoids in a general symmetric monoidal 2-category.

They do more:

Theorem 2.2. The 2-category SMC has all small products and coproducts, and products are strict.

Posted at September 15, 2020 8:41 PM UTC

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11 Comments & 0 Trackbacks

Re: Symmetric Pseudomonoids

It looks to me like the part of their proof dealing with the coproduct structure is purely formal, if interpreted in the internal logic of a symmetric monoidal 2-category.

Posted by: Mike Shulman on September 16, 2020 7:23 PM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

Oh, good. Maybe we can weasel our way out of writing a detailed proof of this one.

Posted by: John Baez on September 16, 2020 7:51 PM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

This result seems to date back to here:

  • Thomas Fox, Coalgebras and Cartesian categories, Comm. Alg. 4 (1976), 665–667.

Dually, the category of commutative monoid objects in a symmetric monoidal category is cocartesian. This was proved in Fox’s suspiciously similar paper in Cocomm. Coalg.

:-)

Posted by: Tom Leinster on September 16, 2020 8:24 PM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

That’s Cocommunications in Coalgebra.

Posted by: John Baez on September 16, 2020 8:29 PM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

Don’t you mean Munnications in Coalgebras?

Posted by: Anon on September 16, 2020 8:42 PM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

Ah, journal abbreviations. What a weird world we make for ourselves, setting up elaborate systems of abbreviation rules in order to save a tiny number of characters, forcing every writer of a reference list to encode the journal names and every reader to decode them again. Today I learned that in the AMS’s abbreviation system, Ann. Phys. means Annalen der Physik, while Ann. Physics means Annals of Physics. Fantastic.

Actually, I’d assumed that the “Comm.” of Comm. Alg. meant commutative, not communications. It’s entirely reasonable that there’d be a journal called Commutative Algebra.

The AMS standard abbreviation for that journal is Comm. Algebra. I think they chose the wrong word to abbreviate: Commutative Alg. would be clearer. Or if they were hell-bent on abbreviating the first word too, they could have gone for Commutat. Alg. Yes, it would be absurd to truncate the word a mere three letters from the end, but not quite as absurd as the fate of the Electronic Journal of Combinatorics, whose first word is abbreviated to Electron. (two letters saved, one punctuation mark added).

Now that I have the list of abbreviations in front of me, I’m finding more and more gems, of which my current favourite is Electromagn. Wirel. Radar Microw. What a saving that is.

I’ll stop now, before I use up all the characters that the journal abbreviators have saved.

Posted by: Tom Leinster on September 16, 2020 10:44 PM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

Tom wrote:

Actually, I’d assumed that the “Comm.” of Comm. Alg. meant commutative, not communications. It’s entirely reasonable that there’d be a journal called Commutative Algebra.

Yes! My joke was designed to exploit precisely this misunderstanding. Sneaky, aren’t I?

Why didn’t anyone notice that “Comm.” means “commutative” in mathematics? I bet the International Organization for Standardization is to blame:

ISO 4 (Information and documentation – Rules for the abbreviation of title words and titles of publications) is an international standard which defines a uniform system for the abbreviation of serial publication titles, i.e., titles of publications such as scientific journals that are published in regular installments.

The International Organization for Standardization (ISO) has appointed the ISSN International Centre as the registration authority for ISO 4. It maintains the List of Title Word Abbreviations (LTWA), which contains standard abbreviations for words commonly found in serial titles. As of August 2017, the standard’s most recent update came in 1997, when its third edition was released.

A major use of ISO 4 is to abbreviate the names of scientific journals using the LTWA. For instance, under ISO 4 standards, the Journal of Biological Chemistry is cited as J. Biol. Chem., and the Journal of Polymer Science Part A should be cited as J. Polym. Sci. A (capitalization is not specified by the standard). The standard notes that “Full stops shall only be used to indicate an abbreviation. Full stops may be omitted from abbreviated words in applications that require limited use of punctuation” (section 4.6)

If I weren’t so busy doing math, I’d love nothing better than to join the International Organization for Standardization. I could spend my days flying around the world, standardizing things.

You may wonder why these experts chose “ISO” as the acronym for “International Organization for Standardization”. They have a reply ready:

Because ‘International Organization for Standardization’ would have different acronyms in different languages (IOS in English, OIN in French), our founders decided to give it the short form ISO. ISO is derived from the Greek word isos (ίσος, meaning “equal”). Whatever the country, whatever the language, the short form of our name is always ISO.

Posted by: John Baez on September 17, 2020 12:06 AM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

In the unlikely event that I ever want to annoy an ISO functionary, I will refer to it as the International Organisation for Standardization.

Posted by: Tom Leinster on September 17, 2020 12:38 AM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

I think you’re giving somebody too much credit for having reasons behind their decisions. One of the abbreviations that’s always annoyed me is Electron. Commun. Probab. for Electronic Communications in Probability. I mean, why even bother abbreviating Electronic to Electron. for a savings of one character? But there’s clearly limited consistency in this system.

Posted by: Mark Meckes on September 17, 2020 7:33 AM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

This result is proved in Theorem 5.2 of

SCHÄPPI, D. (2014). Ind-abelian categories and quasi-coherent sheaves. Mathematical Proceedings of the Cambridge Philosophical Society, 157(3), 391-423. doi:10.1017/S0305004114000401

The proof appears in Appendix A.

Posted by: Daniel Schaeppi on September 17, 2020 12:14 PM | Permalink | Reply to this

Re: Symmetric Pseudomonoids

WOW! This is GREAT!

Thanks so much for proving this, and for noticing my question here.

Posted by: John Baez on September 17, 2020 3:28 PM | Permalink | Reply to this

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