A colax-distributive rig category is like a rig category, but where the distributive laws hold only laxly.
A right colax-distributive rig category is a category $C$ equipped with two monoidal structures $(C,\otimes,I)$ and $(C,\oplus,J)$ and natural transformations
and
making the functor $(-\otimes Z)$ colax monoidal with respect to $\oplus$, and perhaps some other coherence equations. A left colax-distributive rig category has instead transformations
and
making $(Z\otimes -)$ colax monoidal, and a simply colax-distributive rig category has both.
Of course, a rig category is a colax-distributive rig category where the distributivity transformations are isomorphisms.
If $(C,\otimes,I)$ is any monoidal category with finite products, then it becomes a colax-distributive rig category with $(C,\oplus,J)$ the cartesian monoidal structure and the distributivity transformations being induced by the universal property of products.
If $(D,\diamond,\star)$ is a duoidal category in which $\star$ is compatibly braided (or perhaps symmetric), then the category $CComon_\star(D)$ of cocommutative comonoids in $D$ inherits a monoidal structure from $\diamond$ and has finite products; hence it is a colax-distributive rig category.
In a right colax-distributive rig category $(C,\otimes,I,\oplus,J)$, a near-rig is an object $R$ equipped with an $\oplus$-monoid structure $add:R\oplus R \to R$ called “addition”, and an $\otimes$-monoid structure $mult : R\otimes R \to R$ called “multiplication”, such that the right distributive law:
and the right absorption law:
hold.
If also the left distributive and absorption laws hold (which requires $C$ to also be left colax-distributive), then $R$ is a rig. And if $\oplus$ is cartesian and the additive monoidal structure is a group structure, then $R$ is a (near-)ring.
In the right colax-distributive category $CComon_\star(D)$ of cocommutative comonoids in a duoidal category $D$, these agree with the definitions of (cocommutative) (near)-ri(n)gs in $D$ (see duoidal category for these).
On the other hand, if $C$ is a preadditive distributive monoidal category and $\oplus$ is the cartesian monoidal structure (which is also the cocartesian structure, so that $C$ is in fact a rig category), then every object is an $\oplus$-monoid in a unique way, and the distributive and absorption laws are also automatic. Thus, in this case near-rigs and rigs coincide simply with $\otimes$-monoids. If $C$ is additive, then near-rings and rings also coincide with $\otimes$-monoids.
Last revised on April 16, 2020 at 19:55:40. See the history of this page for a list of all contributions to it.