Rig category

In category theory, a rig category (also known as bimonoidal category or 2-rig) is a category equipped with two monoidal structures, one distributing over the other.

Definition
A rig category is given by a category $$\mathbf C$$ equipped with:
 * a symmetric monoidal structure $$(\mathbf C, \oplus, O)$$
 * a monoidal structure $$(\mathbf C, \otimes, I)$$
 * distributing natural isomorphisms: $$\delta_{A,B,C} : A \otimes (B \oplus C) \simeq (A \otimes B) \oplus (A \otimes C)$$ and $$\delta'_{A,B,C} : (A \oplus B) \otimes C \simeq (A \otimes C) \oplus (B \otimes C)$$
 * annihilating (or absorbing) natural isomorphisms: $$a_A : O \otimes A \simeq O$$ and $$a'_A : A \otimes O \simeq O$$

Those structures are required to satisfy a number of coherence conditions.

Examples

 * Set, the category of sets with the disjoint union as $$\oplus$$ and the cartesian product as $$\otimes$$. Such categories where the multiplicative monoidal structure is the categorical product and the additive monoidal structure is the coproduct are called distributive categories.
 * Vect, the category of vector spaces over a field, with the direct sum as $$\oplus$$ and the tensor product as $$\otimes$$.

Strictification
Requiring all isomorphisms involved in the definition of a rig category to be strict does not give a useful definition, as it implies an equality $$A \oplus B = B \oplus A$$ which signals a degenerate structure. However it is possible to turn most of the isomorphisms involved into equalities.

A rig category is semi-strict if the two monoidal structures involved are strict, both of its annihilators are equalities and one of its distributors is an equality. Any rig category is equivalent to a semi-strict one.