Added

- John D. Berman,
*On the commutative algebra of categories*, Algebraic and Geometric Topology, 18(5):2963–3012, August 2018, (arXiv:1606.05606)

and redirects for ’semiring category’ and ’semiring categories’.

]]>fixed the “more generally”-clause

]]>added the category of pointed sets to “Distributive monoidal categories” section

Amy Reed

]]>Added ref to the Johnson-Yau proof of “Baez’s conjecture”.

]]>Just realised that was an abridged version, so have added link to full text.

]]>Added reference

- Niles Johnson, Donald Yau,
*Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory*(arXiv:2107.10526).

Tweaked a couple of references.

]]>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”

]]>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.

Now I have made it read instead like this:

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

distributive category.

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

]]>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).

]]>fix reference to CDH paper (thanks for adding it!)

Antonin Delpeuch

]]>Added

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

- {#Elg20} Josep Elgueta,
*The groupoid of finite sets is biinitial in the 2-category of rig categories*, (arXiv:2004.08684).

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

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

]]>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.)

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

]]>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.

]]>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 $x$.

]]>Also: changed.

]]>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?

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

]]>(And indeed, Laplaza includes them.)

]]>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?

]]>