Rig category
In category theory, a rig category 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 natural isomorphisms: and
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.