Categories of measurement functors. Entropy of discrete amenable group representations on abstract categories. Entropy as a bifunctor into $[0,\infty]$

TL;DR

This paper introduces a unified entropy framework for representations of discrete amenable groups on abstract categories, generalizing classical topological and Kolmogorov-Sinai entropies through measurement functors and bifunctors.

Contribution

It develops a general entropy theory for group representations on abstract categories using measurement functors, unifying classical entropy notions under a categorical framework.

Findings

Entropy decreases along morphisms in the representation category.

Different measurement functors produce entropies that decrease pointwise along natural transformations.

Classical entropies are special cases within this generalized categorical entropy framework.

Abstract

The main purpose of this article is to provide a common generalization of the notions of a topological and Kolmogorov-Sinai entropy for arbitrary representations of discrete amenable groups on objects of (abstract) categories. This is performed by introducing the notion of a measurement functor from the category of representations of a fixed amenable group $\Gamma$ on objects of an abstract category C to the category of representations of $\Gamma$ on distributive lattices with localization. We develop the entropy theory of representations of $\Gamma$ on these lattices, and then define the entropy of a representation of $\Gamma$ on objects of the category C with respect to a given measurement functor. For a fixed measurement functor, this entropy decreases along arrows of the category of representations. For a fixed category, entropies defined via different measurement functors decrease pointwise along natural transformations of measurement functors. We conclude that entropy is a bifunctor to the poset of extended positive reals. As an application of the theory, we show that both topological and Kolmogorov-Sinai entropies are instances of entropies arising from certain measurement functors.