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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 5th 2011

created a simple entry ring object, just for completeness

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeAug 20th 2015
• (edited Aug 20th 2015)

In the article ring object, I actually don’t know what ring operad (suitable for defining rings in general symmetric monoidal categories) is meant. The usual obstacle in defining rings in such general contexts is the distributive law $a(b + c) = a b + a c$, in which the variable $a$ is repeated on one side of the equation; such repetitions need diagonal maps or something similar, which are not available in general symmetric monoidal categories $M$. (There are various dodges, such as passing to cocommutative comonoids in $M$, but let’s put that aside.)

Looking at ring operad, I think something got lost in translation. In that article, there is mention of -rings that come up in e.g. stable homotopy theory, but these are just ordinary monoids with respect to smash products of spectra or of S-modules (in one way of setting things up), where S-modules are already analogous to abelian groups or $\mathbb{Z}$-modules. So there the “ring operad” seems to be just the vanilla monoid operad, but within a conceptual context where we are interpreting the monoid operad in an already sufficiently rich abelian-group-like context so that it makes sense to call their algebras in that context “rings”.

Anyway, unless something is being said that I’m not understanding, I think the sentence at ring object mentioning a “ring operad” needs amendment.

Oh, by the way: I added to ring object the description in terms of Lawvere theories.

• CommentRowNumber3.
• CommentAuthorZhen Lin
• CommentTimeAug 20th 2015

The phrasing of the sentence is strange. One should speak of algebras over an operad, not modules. And the only way I know of defining rings as algebras over an operad is as algebras in $\mathbf{Ab}$ over the associative operad. One could also define commutative rings as algebras over the commutative operad.

I guess the point is that for spectra, there is a whole sequence of operads between the associative operad and the commutative operad. Maybe the “anonymous coward” can explain?

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeAug 20th 2015

I think I’ve heard “module over an operad” more and more. I don’t think it is intrinsically bad terminology, but I prefer the traditional “algebra” mainly because there is little chance of misunderstanding; meanwhile “module” has another meaning in operad theory (a module over an algebra over an operad).

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 20th 2015
• (edited Aug 20th 2015)

@Zhen Lin, I doubt that it helps to wait for the anonymous coward to come by and explain. Just go ahead and improve the passage, given that you understand the matter. If you do feel you need to cross-check with somebody who has a real chance to show up here, try Zoran (who added the sentence in question in rev#2) or Toby (who added the pointer to “ring operad” in rev #4).

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeAug 20th 2015

So there the “ring operad” seems to be just the vanilla monoid operad, but within a conceptual context where we are interpreting the monoid operad in an already sufficiently rich abelian-group-like context so that it makes sense to call their algebras in that context “rings”.

I think this is fairly common in stable homotopy theory.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeAug 20th 2015

Re #6: right. But (as you probably realize) my point was that to the best of my knowledge there is no “ring operad” for use in general symmetric monoidal categories, referring to a statement at ring object, and that the pointer to ring operad there doesn’t explain it away.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeAug 20th 2015

Right.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeAug 20th 2015

Eh, well, I tried to cobble something together at ring object. Experts, please feel free to amend whatever got said in the Idea section.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeAug 21st 2015

Looks good to me!

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeAug 21st 2015

Thanks. I have added a bunch of hyperlinks to the keywords at ring object and I have edited ring operad to at least serve as a redirect to the actual entries that we have on the topic.

• CommentRowNumber12.
• CommentAuthorTodd_Trimble
• CommentTimeAug 21st 2015

Well, thanks to you too, Urs.

There’s still the matter of the page title “ring operad”, which I continue to think is not at all good (or does this phrase appear in the literature? I mean besides in a context where operad could mean cartesian operad = Lawvere theory). If it’s in the literature, I guess we should record it as such (with pinched nose (-: ), but perhaps with some disclaimer. Notice that the text of ring operad so far doesn’t even mention the word ’operad’.

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeAug 21st 2015

I’ve never seen it.

• CommentRowNumber14.
• CommentAuthorTobyBartels
• CommentTimeAug 21st 2015

Google gives me one legitimate hit for ‘ring operad’ outside of the nLab, which is #5 at http://math.umn.edu/~tlawson/hovey/model.html. Perhaps we could ask Mark Hovey what it means.

1. I noticed something interesting: the definition of ring object we have in the nLab is phrased for Cartesian monoidal categories, but the microcosm principle states that the most natural setting for those would be a semiring category, and indeed there’s a very simple definition in this case, which recovers semirings and rings as the semiring objects in $(\mathsf{CMon},\oplus,\otimes_\mathbb{N},0,\mathbb{N})$ and $(\mathsf{Ab}, \oplus,\otimes_\mathbb{Z},0,\mathbb{Z})$.

Has this notion been studied before in the literature?

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeSep 2nd 2021

Just to say that the Idea-section of the entry (here) does mention other forms of defining ring objects, a little closer to what you are suggesting.

Generally, this entry is hardly comprehensive, nor meant to be at this stage. I have added sub-section headers to make clearer that there are different possible definition.

A subsection template “Via the microcosm principle” is now here. That would be a great place to stably host your definition on the MathOverflow page!

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeSep 2nd 2021

There are variant categorifications at 2-rig. Would the choice of one of these make any difference?

2. Added definition of a semiring in a bimonoidal category.

Thanks, Urs! That’s a great idea! :)

I’ve edited the page to include it.

3. There are variant categorifications at 2-rig. Would the choice of one of these make any difference?

Hi David! I think one can define semirings in most of these, not least because it makes sense to consider semi/rings in an arbitrary monoidal category (so e.g. the same definition works for the “monoidal abelian category” or “presentable monoidal category with a cocontinuous tensor product” variant notions of a $2$-rig). However, the definitions tend to be quite a bit more contrived.

Indeed, there are two ways (that I know of) to do this:

1. We can, if the monoidal category is nice enough, put a universal tensor product $\boxtimes$ in $\mathsf{CMon}(\mathcal{C})$ or $\mathsf{Ab}(\mathcal{C})$ and then consider $\boxtimes$-monoids. For $\mathcal{C}=\mathsf{Sets}$, the tensor product $\boxtimes$ becomes either that of commutative monoids, or that of abelian groups. Another example is given in this paper, which does this for bicommutative Hopf algebras;
2. We can also define it by hand if $\mathcal{C}$ is Cartesian (or pass to $\mathsf{CCoMon}(\mathcal{C})$ first if it isn’t). OTOH, the bilinearity diagrams involved are more complicated/bigger:
• We have two octagon diagrams for the relations $a(b+c)=a b+a c$ and $(a+b)c=a c+b c$, where the duplication of $a$ or $c$ is done via the comultiplication map $\Delta_A\colon A\to A\otimes A$. This in particular requires our ring $A$ in $\mathcal{C}$ to be a comonoid.
• There are also two pentagon diagrams for the relations $0a=a$ and $a0=0$, which use the counit morphism $\epsilon\colon A\to\mathbf{1}_{\mathcal{C}}$ of the comonoid structure of $A$.

For comparison, this definition involves only two pentagons and two squares, and in particular the duplication of $A$ comes via the distributivity map $A\otimes(A\oplus A)\dashrightarrow(A\otimes A)\oplus(A\otimes A)$ of $\mathcal{C}$, so $A$ need not additionally be a comonoid in $\mathcal{C}$.

Secondly, this seems to be the appropriate definition if one regards the microcosm principle as “passing from monoids to pseudomonoids”. For comparison, let me first recall how the usual story goes for for monoids in monoidal categories:

1. First, we start with the notion of a monoid in $(\mathsf{Sets},\times,\mathrm{pt})$, i.e. just a monoid;
2. Categorifying this, we obtain pseudomonoids in $(\mathsf{Cats},\times,\mathsf{pt})$, i.e. monoidal categories;
3. Then, we observe that there’s a natural definition of a monoid in a monoidal category $\mathcal{C}$, which uses the monoidal structure of $\mathcal{C}$ to formulate the monoid axioms. For example, the associator of $\mathcal{C}$ enters in the formulation of the associativity condition for monoids in $\mathcal{C}$.

Then, we have an exactly analogous pattern for “semirings in bimonoidal categories”:

1. First, we start with the notion of a semiring in $(\mathsf{CMon},\oplus,\otimes_{\mathbb{N}},0,\mathbb{N})$, which gives just a semiring;
2. Categorifying this, we obtain pseudomonoids in $(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})$, which are bimonoidal categories (up to having 3 less axioms than Laplaza’s definition);
3. Then, we observe that there’s a natural definition of a semiring in a bimonoidal category $\mathcal{C}$, which uses the bimonoidal structure of $\mathcal{C}$ to formulate the semiring axioms. For example, the left distributor morphism $\delta^{\ell}_{R,R,R}\colon R\otimes(R\oplus R)\to(R\otimes R)\oplus(R\otimes R)$ is used in formulating one of the four bilinearity conditions for $\mathcal{C}$.

So in short we have a kind of “table of analogies”:

• set | category
• commutative monoid | symmetric monoidal category
• monoid in commutative monoids = semiring | pseudomonoid in symmetric monoidal categories = bimonoidal category
• monoids in monoidal categories | semirings in bimonoidal categories
• axioms for monoids in monoidal categories use the monoidal structure | axioms for semirings in bimonoidal categories use the bimonoidal structure

(By the way, sorry for the long reply!)