  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.

Dynamical Systems