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concerning the discussion here: notice that an entry rig category had once been created, already.
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?
(And indeed, Laplaza includes them.)
It’s worth noting though that in the case where $\oplus$ is the categorical coproduct, that $0 \otimes x \cong 0$ comes for free.
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?
Also: changed.
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 $x$.
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.
I thought my reasoning worked (where $y = x$) since $\hom(x, x)$ is inhabited. No?
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 added to distributive monoidal category a modified claim (remark 1) and proof.
Okay, great. I moved this remark out of the “Definition” section where I didn’t think it exactly belonged.
Added
A string diagram treatment of rig categories via sheet diagrams is in
Biinitiality of the groupoid of finite sets is shown in
Don’t we have any mentioning of homomorphisms of rig categories, under any name?
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.
Gave the publication details for
which establishes what is called “Baez’s conjecture”
Added reference
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