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 $\mathbf {C}$ equipped with:

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

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

Examples

Set, the category of sets with the disjoint union as $\oplus$ 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.^[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

^ ^a ^b 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.
^ 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.
^ 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

[laplaza-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.

[kelly-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.

[guillou-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.

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