Category theoretic properties of the A. R\'enyi and C. Tsallis entropies

TL;DR

This paper embeds Rényi and Tsallis entropies into a categorical framework, establishing their algebraic properties and axiomatic foundations within a specially constructed category of measured spaces.

Contribution

It introduces a categorical formalism for Rényi and Tsallis entropies, demonstrating their algebraic compatibility and axiomatic generalization in the category MES.

Findings

Both entropies are compatible with categorical operations.

The functional defining the entropies is additive and multiplicative.

A universal exponent parameterizes the entropies across the category.

Abstract

The problem of embedding the Tsallis and R\'{e}nyi entropies in the framework of category theory and their axiomatic foundation is studied. To this end, we construct a special category MES related to measured spaces. We prove that both of the R\'{e}nyi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic functional that appears in their definitions, as well as in the associated Lebesgue space norms, has good algebraic compatibility properties. We prove that this functional is both additive and multiplicative with respect to the direct product and the disjoint sum (the coproduct) in the category MES, so it is a natural candidate for the measure of information or uncertainty. We prove that the category MES can be extended to monoidal category, both with respect to the direct product as well as to the coproduct. The basic axioms of the original R\'{e}nyi entropy theory are generalized and reformulated in the framework of category MES and we prove that these axioms foresee the existence of an universal exponent having the same values for all the objects of the category MES. In addition, this universal exponent is the parameter, which appears in the definition of the Tsallis and R\'{e}nyi entropies.