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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 3rd 2010
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeSep 10th 2012

at bimonoidal category I added references, and briefly recorded the strictification results, mentioned the basic examples.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeSep 10th 2012

I’m not familiar with the term “bimonoidal”. But the naming seems very far from optimal.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 10th 2012

This is the standard term in the literature, as far as I am aware. Most of the interest in these categories is in the context of algebraic K-theory, see at K-theory of a bipermutative category.

• CommentRowNumber5.
• CommentAuthorTodd_Trimble
• CommentTimeSep 10th 2012

Thanks. It’s possible that I’ve simply forgotten I’d heard of it.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeSep 10th 2012

It’s pretty awful, isn’t it? We should call them rig categories.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeSep 10th 2012

Mike, I would love making some such suggestion in the nLab article. “Rig category” seems pretty good to me.

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeSep 10th 2012

If you really want to be systematic, you should call them rigdal categories. Or else say monoid category for monoidal category ;-)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeSep 10th 2012
• (edited Sep 10th 2012)

Also, there is no condition that a bimonoial category be abelian or similar. So for the general definition “rig” seems inappropriate and “(bi-monoid)-al category” indeed more appropriate.

Of course some of the standard examples like $R Mod$ etc are indeed 2-rigs. Peter May in his book has a nice list of examples of a very different form.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeSep 10th 2012

Does May have examples where not even the additive monoidal structure $\oplus$ is symmetric?

• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeSep 10th 2012

The problem with “bimonoid(al)”, by the way, is that it gives no hint as to how the two monoidal structures are related. In the absence of any such hint, my default assumption would tend to be that they are related in some symmetrical way, rather than that one of them distributes over the other. It also clashes quite badly with bialgebra.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 10th 2012

No I think it’s always symmetric, that’s true.

• CommentRowNumber13.
• CommentAuthorTodd_Trimble
• CommentTimeSep 11th 2012

Also, there is no condition that a bimonoial category be abelian or similar. So for the general definition “rig” seems inappropriate and “(bi-monoid)-al category” indeed more appropriate.

No, it’s clear that it’s not more appropriate, for exactly the reasons Mike gives in #11 (which were exactly my thoughts). Besides that, there is no reason I know of that “rig category” should force the the $\oplus$ operation to be a biproduct; therefore I don’t see that #9 applies. “Rig category” seems far more appropriate to me.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeSep 11th 2012

No

Sure.

• CommentRowNumber15.
• CommentAuthorMike Shulman
• CommentTimeSep 11th 2012

So if a rig is a ring without Negatives, and a rng is a ring without an Identity, what is a ring whose addition isn’t necessarily commutative?

• CommentRowNumber16.
• CommentAuthorZhen Lin
• CommentTimeSep 11th 2012

Following the pattern, it would have to be either a “rin” or an “ing”. We could say r stands for “reversible addition”… But what is such a thing, really? Distributivity implies commutative addition in the presence of a one-sided multiplicative unit and two-sided cancellation.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeSep 11th 2012
• (edited Sep 11th 2012)

It is not clear to me that dropping letters on the word “ring” is a more sober step than a somewhat unspecific use of the prefix “bi”. One shouldn’t take one’s puns too seriously, fun as they may be at the beginning.

Certainly at the conferences where I usually go to, it’s the former that raises eyebrows, not the latter. At least when we get to “rng” a certain boundary is being crossed that shouldn’t be crossed.

I think if you really want to improve on “bimonoidal”, then a serious suggestion for a new term is called for. Maybe something involving “distrib—”, because that will automatically imply two monoidal structures and a compatibility.

• CommentRowNumber18.
• CommentAuthorMike Shulman
• CommentTimeSep 11th 2012

I think my inclination would be to write at bimonoidal category something like

A bimonoidal category is a term commonly used among algebraic topologists for a rig category whose additive monoidal structure may not be symmetric. Since this usage clashes with bimonoid, and all interesting examples do have symmetric addition, on the nLab we generally speak of rig categories instead of bimonoidal ones.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeJan 28th 2022

For when the editing functionality is back, to add pointer to this reference:

• Niles Johnson, Donald Yau, Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory, 1409 pages (web, pdf, front+back-matter+appendix: arXiv:2107.10526 (127 pages))