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 ${\displaystyle \mathbf {C} }$ 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]}

Examples

• 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 ${\displaystyle \otimes }$.

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

References

1. Kelly, G. M. (1974). "Coherence theorems for lax algebras and for distributive laws". Category Seminar. Lecture Notes in Mathematics. Vol. 420. pp. 281–375. doi:10.1007/BFb0063106. ISBN   978-3-540-37270-7.
2. Laplaza, Miguel L. (1972). "Coherence for distributivity" (PDF). In G. M. Kelly; M. Laplaza; G. Lewis; Saunders Mac Lane (eds.). Coherence in Categories. Lecture Notes in Mathematics. Vol. 281. Springer Berlin Heidelberg. pp. 29–65. doi:10.1007/BFb0059555. ISBN   978-3-540-05963-9 . Retrieved 2020-01-15.
3. Guillou, Bertrand (2010). "Strictification of categories weakly enriched in symmetric monoidal categories". Theory and Applications of Categories. 24 (20): 564–579. arXiv: 0909.5270 .

• Rig category at the nLab