Rig category

A rig category is given by a category ${\displaystyle \mathbf {C} }$ equipped with:

• a symmetric monoidal structure ${\displaystyle (\mathbf {C} ,\oplus ,O)}$
• a monoidal structure ${\displaystyle (\mathbf {C} ,\otimes ,I)}$
• distributing natural isomorphisms: ${\displaystyle \delta _{A,B,C}:A\otimes (B\oplus C)\simeq (A\otimes B)\oplus (A\otimes C)}$ and ${\displaystyle \delta '_{A,B,C}:(A\oplus B)\otimes C\simeq (A\otimes C)\oplus (B\otimes C)}$
• annihilating (or absorbing) natural isomorphisms: ${\displaystyle a_{A}:O\otimes A\simeq O}$ and ${\displaystyle a'_{A}:A\otimes O\simeq O}$

Those structures are required to satisfy a number of coherence conditions.[1][2]

• Set, the category of sets with the disjoint union as ${\displaystyle \oplus }$ and the cartesian product as ${\displaystyle \otimes }$. 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 ${\displaystyle \oplus }$ and the tensor product as ${\displaystyle \otimes }$.

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 ${\displaystyle A\oplus B=B\oplus A}$ 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]

• Rig category at the nLab