  Given any finite set equipped with a probability measure, one may compute its
Shannon entropy or information content. The entropy becomes the logarithm of
the cardinality of the set when the uniform probability is used. Leinster
introduced a notion of Euler characteristic for certain finite categories, also
known as magnitude, that can be seen as a categorical generalization of
cardinality. This paper aims to connect the two ideas by considering the
extension of Shannon entropy to finite categories endowed with probability, in
such a way that the magnitude is recovered when a certain choice of "uniform"
probability is made.
