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.
A rig category is given by a category equipped with:
Those structures are required to satisfy a number of coherence conditions. [1] [2]
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. [1]
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. [3]
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.
A rig category is given by a category equipped with:
Those structures are required to satisfy a number of coherence conditions. [1] [2]
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. [1]
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. [3]