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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 10th 2012

concerning the discussion here: notice that an entry rig category had once been created, already.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeJul 17th 2013

In a ring, the absorption/annihilation law $0\cdot x = 0$ follows from distributivity and additive inverses, since $0\cdot x + 0\cdot x = (0+0)\cdot x = 0\cdot x$ and we can cancel one copy to obtain $0\cdot x = 0$. In a rig, however, we have to assert absorption separately. So shouldn’t a rig category also include isomorphisms $x\otimes 0 \cong x \cong 0\otimes x$ as part of its structure?

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeJul 17th 2013

(And indeed, Laplaza includes them.)

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJul 17th 2013

It’s worth noting though that in the case where $\oplus$ is the categorical coproduct, that $0 \otimes x \cong 0$ comes for free.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeJul 17th 2013

Are you saying that if we assume $\otimes$ preserves binary coproducts in each variable, then it automatically preserves initial objects in each variable as well?

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJul 17th 2013

Also: changed.

• CommentRowNumber7.
• CommentAuthorTodd_Trimble
• CommentTimeJul 17th 2013

Yes: if we assume the natural canonical map $x \otimes y + x \otimes z \to x \otimes (y + z)$ is an isomorphism, then putting $y = 0$, $z = 1$ (the monoidal unit), we obtain a natural isomorphism

$x \otimes 0 + x \to x$

whose restriction to the inclusion of $x$ is the identity $x \to x$. Let $k$ be its restriction to the inclusion of $x \otimes 0$. Then we obtain a bijection

$\hom(x, x) \stackrel{\langle 'k', id \rangle}{\to} \hom(x \otimes 0, x) \times \hom(x, x)$

and this forces $\hom(x \otimes 0, x)$ to be a singleton, for any $x$.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJul 18th 2013

Did you mean to say that we get a bijection

$\hom(x, y) \stackrel{\langle 'k', id \rangle}{\to} \hom(x \otimes 0, y) \times \hom(x, y)$

for any $x$ and $y$, hence $hom(x\otimes 0,y)$ is a singleton? Anyway, this is nice — it should go at distributive monoidal category.

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeJul 18th 2013

I thought my reasoning worked (where $y = x$) since $\hom(x, x)$ is inhabited. No?

• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeJul 19th 2013
• (edited Jul 19th 2013)

My apologies: I see what Mike is saying, and now I don’t see how to prove my claim. (What happened is that I remembered a (true) statement that the claim holds in the case of cartesian monoidal categories, and thought that would generalize right away to the more general case.)

(And, in fact, it’s trivially false. For example, the coproduct $\vee$ on a join-semilattice preserves binary coproducts in each variable but not the initial object.)

• CommentRowNumber11.
• CommentAuthorTodd_Trimble
• CommentTimeJul 19th 2013

I added to distributive monoidal category a modified claim (remark 1) and proof.

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeJul 20th 2013

Okay, great. I moved this remark out of the “Definition” section where I didn’t think it exactly belonged.

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeOct 27th 2020

A string diagram treatment of rig categories via sheet diagrams is in

• {#CDH} Cole Comfort, Antonin Delpeuch, Jules Hedges, Sheet diagrams for bimonoidal categories, (arXiv:2010.13361)

Biinitiality of the groupoid of finite sets is shown in

1. fix reference to CDH paper (thanks for adding it!)

Antonin Delpeuch

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeJan 8th 2021

I have added brief mentioning of the example of the distributive category of topological spaces, and of the distributive monoidal category of pointed topological spaces (with respect to wedge sum and smash product).

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeJan 8th 2021

Don’t we have any mentioning of homomorphisms of rig categories, under any name?

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeJan 8th 2021

I have rephrased this paragraph

If $\oplus$ is the category-theoretic coproduct and $\otimes$ is the category-theoretic product (Cartesian product), then we have a distributive category, which is a special case of a rig category.

because it made it sound as if taking the coproduct and product always yields a distributive category.

A rig category where $\oplus$ is the category-theoretic coproduct and $\otimes$ is the category-theoretic product (Cartesian product) is called a distributive category.

• CommentRowNumber18.
• CommentAuthorDavidRoberts
• CommentTimeMay 6th 2021

Gave the publication details for

• {#Elg21} Josep Elgueta, The groupoid of finite sets is biinitial in the 2-category of rig categories, Journal of Pure and Applied Algebra 225 Issue 11 (2021) 106738, doi:10.1016/j.jpaa.2021.106738, arXiv:2004.08684).

which establishes what is called “Baez’s conjecture”

• CommentRowNumber19.
• CommentAuthorDavidRoberts
• CommentTimeMay 6th 2021

Tweaked a couple of references.

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeJul 23rd 2021