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Rig category

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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

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A rig category is given by a category equipped with:

  • a symmetric monoidal structure
  • a monoidal structure
  • distributing natural isomorphisms: and
  • annihilating (or absorbing) natural isomorphisms: and

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

Examples

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  • Set, the category of sets with the disjoint union as and the cartesian product as . 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 and the tensor product as .

Strictification

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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]

References

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  1. ^ 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.
  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.