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The main purpose of this article is to provide a common generalization of thenotions of a topological and Kolmogorov-Sinai entropy for arbitraryrepresentations of discrete amenable groups on objects of (abstract)categories. This is performed by introducing the notion of a measurementfunctor 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 entropytheory of representations of $\Gamma$ on these lattices, and then define theentropy of a representation of $\Gamma$ on objects of the category C withrespect to a given measurement functor. For a fixed measurement functor, thisentropy decreases along arrows of the category of representations. For a fixedcategory, entropies defined via different measurement functors decreasepointwise along natural transformations of measurement functors. We concludethat entropy is a bifunctor to the poset of extended positive reals. As anapplication of the theory, we show that both topological and Kolmogorov-Sinaientropies are instances of entropies arising from certain measurement functors.

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