A Characterization of Entropy as a Universal Monoidal Natural Transformation

Abstract: We show that the essential properties of entropy (monotonicity, additivity and subadditivity) are consequences of entropy being a monoidal natural transformation from the under category functor $-/\mathsf{LProb}{\rho}$ (where $\mathsf{LProb}{\rho}$ is category of $\rho$-th-power-summable probability distributions, $0<\rho<1$) to $\Delta_{\mathbb{R}}$. Moreover, the Shannon entropy can be characterized as the universal monoidal natural transformation from $-/\mathsf{LProb}{\rho}$ to the category of integrally closed partially ordered abelian groups (a reflective subcategory of the lax-slice 2-category over $\mathsf{MonCat}{\ell}$ in the 2-category of monoidal categories), providing a succinct characterization of Shannon entropy as a reflection arrow. We can likewise define entropy for every monoidal category with a monoidal structure on its under categories (e.g. the category of finite abelian groups, the category of finite inhabited sets, the category of finite dimensional vector spaces, and the augmented simplex category) via the reflection arrow. This implies that all these entropies over different categories are components of a single natural transformation (the unit of the idempotent monad), allowing us to connect these entropies in a natural manner. We also provide a universal characterization of the conditional Shannon entropy based on the chain rule which, unlike the characterization of information loss by Baez, Fritz and Leinster, does not require any continuity assumption.