nLab colax-distributive rig category

Idea

A colax-distributive rig category is like a rig category, but where the distributive laws hold only laxly.

Definition

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

$$(X\oplus Y) \otimes Z \to (X\otimes Z) \oplus (Y\otimes Z)$$

and

$$J \otimes Z \to J$$

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

$$Z \otimes (X\oplus Y) \to (Z\otimes X) \oplus (Z\otimes Y)$$

and

$$Z \otimes J \to J$$

making $(Z\otimes -)$ colax monoidal, and a simply colax-distributive rig category has both.

Examples

• 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.

Structures in colax-distributive rig categories

Rings and near-rings

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:

$$\array{ (R\oplus R) \otimes R & \xrightarrow{add\otimes id} & R\otimes R & \xrightarrow{mult} & R\\ \downarrow & & & & \uparrow^{add}\\ (R\otimes R) \oplus (R\otimes R) && \xrightarrow{mult \oplus mult} && R\oplus R }$$

and the right absorption law:

$$\array{ J\otimes R & \to & J & \xrightarrow{0} & R\\ & _{0\otimes id}\searrow && \nearrow_\mult \\ && R\otimes R }$$

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.

Related pages

• distributivity for monoidal structures