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

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.