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.
References.
- Introduction to Categorical Quantum Mechanics (Heunen, Vicary)
- Coalgebras and Cartesian Categories (Fox)
- Graphical Linear Algebra (Sobocinski, blog posts)
Minimal Fox’s theorem
References
Instances of Fox’s theorem.
See: Fox’s theorem.