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  equipped with:
 a symmetric monoidal structure  
 a monoidal structure 
  distributing natural isomorphisms:  and 
 annihilating (or absorbing) natural isomorphisms:  and 

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

 Set, the category of sets with the disjoint union as  and the cartesian product as . 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  and the tensor product as . 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  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. References

 

Monoidal categories