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 $𝐂$ equipped with:

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

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.

References

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