nLab Fox's theorem

Redirected from "Fox theorem".
Note: Fox's theorem and Fox's theorem both redirect for "Fox theorem".
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Contents

Idea

This is a theorem due to Thomas Fox which characterizes products in a cartesian monoidal category as an algebraic structure given by natural transformations rather than in terms of a universal poperty.

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.

Reference

Last revised on July 16, 2024 at 19:01:18. See the history of this page for a list of all contributions to it.