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.
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 August 18, 2025 at 09:27:04. See the history of this page for a list of all contributions to it.