nLab
colax-distributive rig category

Colax-distributive rig categories

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 CC equipped with two monoidal structures (C,,I)(C,\otimes,I) and (C,,J)(C,\oplus,J) and natural transformations

(XY)Z(XZ)(YZ) (X\oplus Y) \otimes Z \to (X\otimes Z) \oplus (Y\otimes Z)

and

JZJ J \otimes Z \to J

making the functor (Z)(-\otimes Z) colax monoidal with respect to \oplus, and perhaps some other coherence equations. A left colax-distributive rig category has instead transformations

Z(XY)(ZX)(ZX) Z \otimes (X\oplus Y) \to (Z\otimes X) \oplus (Z\otimes X)

and

ZJJ Z \otimes J \to J

making (Z)(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,,I)(C,\otimes,I) is any monoidal category with finite products, then it becomes a colax-distributive rig category with (C,,J)(C,\oplus,J) the cartesian monoidal structure and the distributivity transformations being induced by the universal property of products.

  • If (D,,)(D,\diamond,\star) is a duoidal category in which \star is compatibly braided (or perhaps symmetric), then the category CComon (D)CComon_\star(D) of cocommutative comonoids in DD 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,,I,,J)(C,\otimes,I,\oplus,J), a near-rig is an object RR equipped with an \oplus-monoid structure add:RRRadd:R\oplus R \to R called “addition”, and an \otimes-monoid structure mult:RRRmult : R\otimes R \to R called “multiplication”, such that the right distributive law:

(RR)R addid RR mult R add (RR)(RR) multmult RR \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:

JR J 0 R 0id mult RR \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 CC to also be left colax-distributive), then RR is a rig. And if \oplus is cartesian and the additive monoidal structure is a group structure, then RR is a (near-)ring.

In the right colax-distributive category CComon (D)CComon_\star(D) of cocommutative comonoids in a duoidal category DD, these agree with the definitions of (cocommutative) (near)-ri(n)gs in DD (see duoidal category for these).

On the other hand, if CC is a preadditive distributive monoidal category and \oplus is the cartesian monoidal structure (which is also the cocartesian structure, so that CC 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 CC is additive, then near-rings and rings also coincide with \otimes-monoids.

Last revised on July 17, 2013 at 20:52:15. See the history of this page for a list of all contributions to it.