With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A theorem due to Fox 1976 characterizes the cartesian products in a cartesian monoidal category as an algebraic structure given by natural transformations rather than in terms of a universal property.
Fox’s original theorem is as follows.
The forgetful functor from the category of cartesian monoidal categories and strict monoidal functors into the category of symmetric monoidal categories and strict monoidal functors admits a right adjoint, given by the construction of the category of cocommutative comonoids.
(Note that the forgetful functor also has a left adjoint, by two-dimensional monad theory.)
However, the following corollary (and variants) are often also referred to simply as “Fox’s theorem”:
A symmetric monoidal category is cartesian if and only if it is isomorphic to its own category of cocommutative comonoids. Thus every object is equipped with a unique cocommutative comonoid structure, and these structures are respected by all maps.
The original article:
A survey of theorems extending Fox’s original theorem:
Last revised on September 18, 2025 at 11:07:17. See the history of this page for a list of all contributions to it.