Mario Román

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instances of Fox's theorem

Last updated Jul 23, 2024

Fox’s theorem comes in multiple versions: two of them describe the construction of the cofree and the free cartesian category over a monoidal category. The more compact formulation says that any symmetric monoidal category is a cartesian category exactly if each object is a cocommutative comonoid and each morphism is a comonoid homomorphism, in a natural and uniform way.

The adjunctions of Fox’s theorem are hom-tensor adjunctions between the tensor of monoidal categories and the exponential of symmetric monoidal categories. These explain the requirements for naturality and uniformity in Fox’s theorem.

structural-foxs-theorem

References

Instances of Fox’s theorem.

instances-of-fox-theorem

See: Fox’s theorem.