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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 follows from distributivity and additive inverses, since and we can cancel one copy to obtain . In a rig, however, we have to assert absorption separately. So shouldn’t a rig category also include isomorphisms as part of its structure?
(And indeed, Laplaza includes them.)
It’s worth noting though that in the case where is the categorical coproduct, that comes for free.
Are you saying that if we assume 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 is an isomorphism, then putting , (the monoidal unit), we obtain a natural isomorphism
whose restriction to the inclusion of is the identity . Let be its restriction to the inclusion of . Then we obtain a bijection
and this forces to be a singleton, for any .
Did you mean to say that we get a bijection
for any and , hence is a singleton? Anyway, this is nice — it should go at distributive monoidal category.
I thought my reasoning worked (where ) since 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 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 is the category-theoretic coproduct and 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 is the category-theoretic coproduct and 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
Added
and redirects for ’semiring category’ and ’semiring categories’.
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