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The problem of embedding the Tsallis, Rényi and generalized Rényi 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 Rényi and Tsallis entropies can be imbedded in the formalism of category theory by proving that the same basic partition 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 Rényi 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 Rényi entropies. It is proved that in a similar manner, the partition functional that appears in the definition of the Generalized Rényi entropy is a multiplicative functional with respect to direct product and additive with respect to the disjoint sum, but its symmetry group is reduced compared to the case of classical Rényi entropy.

The discovery of two related generalizations of the classical Shannon entropy [

Because the measure of information is a basic scientific concept, in this work we develop a formalism in the framework of the category theory [

The paper is organized as follows. In the Section 2, Subsection 2.1, we define a special category related to measurable spaces (referred to as MES), enabling the introduction an associated basic functional z_{p} (see the forthcoming Section for his exact definition). Both the Tsallis and Rényi entropies, as well as the distance in l_{p} spaces, may be expressed in terms of this functional. In the Subsection 2.2, we define the direct product of the objects in MES and we prove that the functional z_{p} satisfies a compatibility relation with respect to this product i.e., it is multiplicative. This multiplicative property is equivalent to the additivity of the Rényi entropy. In the Subsection 2.3, we define the disjoint sum (or the coproduct) of the objects in MES, and we prove that the functional z_{p} satisfies a compatibility relation with respect to coproduct i.e., it is additive. Note that this property is equivalent to one of the postulates characterizing the Rényi entropy. The proofs that both product and coproduct possess a universal property and that the direct product and coproduct can also be defined for morphisms of the category MES, can be found in the Subsection 2.4. In the Subsection 2.5 we show that, by extending the category MES with the introduction of the unit object and the null object, the category MES becomes to a monoidal category.

Section 3 deals with the axiomatic characterization of the functional z_{p}. We demonstrate that there exists a universal exponent p (the same for all the objects of the category) that characterizes completely the functional z_{p} (hence, also the Tsallis or Rényi entropies) up to an arbitrary multiplicative factor. In Section 4, it is proven that the main properties of the Rényi entropy, which are used in the axiomatic and category theoretic formulation, can be reformulated in order to be generalized to the case of the generalized Rényi entropy (GRE). The symmetry properties of GRE are studied in Subsection 4.1. Appendix 1 shows that the Rényi divergence can be expressed in terms of the Rényi entropy. The proof of the universality (with respect to all the objects of the category MES) of the exponent defining the Rényi or Tsallis entropies can be found in Appendix 2. In Appendix 3 some algebraic results related to the symmetry of GRE are proved.

Our definitions include as a particular case the original definition of the generalized entropies [

Remark 1 At first sight it would be more natural to consider the objects as measure space triplet

We denote with

finite norm (pseudo norm, respectively): more precisely,

for some non-negative density

where

In this framework, for a given measurable space

which in the case of discrete distribution, X a denumerable set,

For a given measurable space

the density ρ in the Banach space for

These relations give the geometrical interpretation of the generalized entropies (for further information Refs to [

Remark 2 The study of the generalized entropies helps us to better understand the classical entropy. For^{p} norm, and for ^{p}-norm [^{p} spaces are reflexive, the Maxent problem is equivalent to the minimal L^{p} distance problem with restrictions [^{p} spaces has, in general, trivial duals, the Maxent problem is equivalent to the maximal L^{p} distance or the maximal

The corresponding generalized entropy

Consider now a measure space

Note that the Rényi divergence [

is related to the Rényi entropies (see Appendix 1). Note that when x is a finite or denumerable set, if we denote with

Remark that, in this particular case,

In the framework of the our formalism, the multiplicative property is the counterpart of the Postulate 4 in the Rényi theory [

Here

Consider now the measures

with

We have the following basic proposition

Proposition 3 Let

Then we have

The validity of this statement follows directly from the definitions of the direct product, the Rényi entropy and the functional z_{p}.

Let us study now the property encoded in the Postulate 5’ related to the Rényi entropy theory (Ref. [

Definition 4 The coproduct of measurable spaces

Here,

Let

We restrict our definition of coproduct to finite terms. An example of (denumerable infinite) coproduct is the grand canonical ensemble.

Remark 5 If

From the previous definition of the direct sum and the functional

Proposition 6 The reformulation of the Postulate 5’ (Ref. [

In the following we prove that the basic binary operations on measurable spaces, the direct product and the direct sum, defined in the previous section, have universality properties in the category of measurable spaces MES.

Consider the direct product

Proposition 7 In the category MES the applications

Proof. The measurability of

From the previous Proposition 7 results immediately the following Theorem

Theorem 8 In the category MES, the direct product has the universal property. Let

where

Proof. The morphism θ is induced by the application

In conclusion the direct product operation has the natural functorial property, so the multiplicative property Equation (24) of the functional

Proposition 9 In the category MES, consider the objects

Proof. The injections

By reversing the arrows, in analogy to the Theorem 8, we obtain the following result.

Theorem 10 In the category mes the direct sum of the objects has the following universality property. Let denote with

where

Proof. The morphism

In conclusion, the direct sum operation has natural category theoretic properties. Hence, the additivity property Equation (29) of the functional

We recall the following

Proposition 11 [

For the pair of morphisms

We denoted with

Let

If in the category

Similarly, by duality arguments, we have the following result for the direct sum (coproduct)

Proposition 12 [

For the pair of morphisms

We denoted with

If in the category

We emphasize that, despite the fact that the construction of the direct sum is dual to the direct product, from the previous proposition (12) the functor G is a covariant functor. In the category mes we have an unit object as well as the null object. The unit object is denoted with

Conclusion 13 The category MES is a monoidal category both with respect to the product

We expose another approach, based on category theory, to the problem of the naturalness of the choice of the family of functions

Note that

Then, the Postulate 1 (the symmetry property) and Postulate 5’ (the additivity property expressed in Propo- sition 6) can be generalized as follows. Postulate 1 & Postulate 5’

for some Borel measurable function

The last requirement result by considering the case when the support of

In our settings, the analog of the Postulate 4 (the additivity property) [

By arguments similar to the proof of the uniqueness, from Theorem 2 [

Remark that all of the definitions of the classical, Rényi, Tsallis entropies contains only set theoretic and mea- sure theoretic concepts, no supposition on the auxiliary algebraic or differentiable structure associated to the measure space are assumed, so their definitions can be used t, continuos or discrete distributions. In the case of discrete measured space the classical definitions of the entropies Equations (7), (13)-(15) are invariant under the permutation group of the elements of the discrete set. This invariance encodes the assumption of complete apriory lack of information about the physical system, this absolute ignorance is lifted by the specification of the probability density function. On the other hand, consider the case when the measure space has the product structure

such that

Suppose that the probability measure on

The GRE’s associated are [

We remark that in the definitions Equation (48), the role of the variables

In the limit case

In order to prove that in the case of the GRE the symmetry group is reduced to some subgroup, we consider only a special case: the spaces

We use the array notation

It is invariant under the transformation (see Lemma 16)

where the transformation

Suppose we are in general case, when the indices i, a has completely different physical interpretation. Its is clear that the measure of information of such a system cannot be invariant under the permutation group

Similarly we are interested to find the subgroup

By using the Corollary 17, we obtain the following conclusion concerning the symmetry group of GRE, com- pared to the symmetry group of the classical Rényi or Tsallis entropies.

Proposition 14 The symmetry group

where

where the map

is the direct product

where

In conclusion, in this particular case, the symmetry group associated to GRE’s

According to Equations (42)-(49), the additivity of the GRE is equivalent to the multiplicative property of the functionals

Under these assumptions and with the notations Equations (47) and (49), we have the following functorial property with respect to the direct product:

It is possible to extend, partially, the additivity property from Proposition 6. Consider the measured space defined in Equations (42)-(46) and suppose that the space X and the related objects has the following decom- position in direct sum, similar to the Definition 4

We define the measure

similar to Definition 4

Under previous conditions Equations (67)-(71), we have the following additivity result:

We obtain a similar result for the functional

We proved that the most natural setting for treating the axiomatic approach to the study of definitions of measures of information or uncertainty, is the formalism of measure spaces and of the category theory. The Rényi divergence can be reduced to the Rényi entropy in our measure theoretic formalism. Category theory was invented for the most difficult, apparently contradictory aspects of the foundation of mathematics. In this respect, we introduced a category of measurable spaces MES. We proved that in the category MES existed the direct product and the direct sum, having universal properties. We proved that the functional

The main conclusions may be summarized as follows:

1) The natural measure of the quantity of information is the family of functionals

2)The category MES is the natural framework for treating the problems related to the measure of the infor- mation, in particular in reformulating the Rényi axioms;

3) The category MES is a monoidal category with respect to direct product and coproduct and the functional

4) Up to a multiplicative constant, it is possible to recover the exact form of the functional

5) The GRE

6) The symmetry group of

7) The Postulate 5'’of the classical Rényi entropy appears in the case of GRE as the additivity property of the functional

The authors are grateful to Prof. M. Van Schoor and Dr D. Van Eester from Royal Military School, Brussels. György Steinbrecher is grateful to Prof. C. P. Niculescu from Mathematics Department, University of Craiova, Romania, and S. Barasch for discussions on category theory. Giorgio Sonnino is also grateful to Prof. P. Nar- done and Dr. P. Peeters of the Université Libre de Bruxelles (ULB) for useful discussions and suggestions.

GyörgySteinbrecher,AlbertoSonnino,GiorgioSonnino,11,11, (2016) Category Theoretic Properties of the A. Rényi and C. Tsallis Entropies. Journal of Modern Physics,07,251-266. doi: 10.4236/jmp.2016.72025

Suppose to have a measurable space

Consider now a measurable space

The Rényi divergence reads

According to the Equations (73, 74, 76) and normalization Equation (75), we get

Using Equation (35) with

Hence, Equation (38) reads

In the particular case

From Equations (81), (82) results

We select in Equation (83)

and the following equation results

Remark t hat putting in Equation (84)

The general solution of corresponding homogenous equation

may be found by using again the continuity of the function

Here

The general solution of the Equation (85) reads

and similarly we have for all of the object of the category mes

By using Equations (81), (89), (90), (91), we get the universal linear slope p

and, by Equations (78)-(80), up to undetermined multiplicative constants

Lemma 16 Let

where

Proof. We proceed by induction. For

By using the previous Lemma 16 in two successive steps, with

Corollary 17 Suppose that for all

where

where the map