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.

翻译：给定任意一个配备概率测度的有限集合，可以计算其香农熵或信息含量。当采用均匀概率时，熵即为集合基数的对数。莱因斯特为某些有限范畴引入了欧拉示性数的概念（亦称幅度），该概念可视为基数概念的范畴化推广。本文旨在通过将香农熵推广到赋予概率的有限范畴，使得在特定"均匀"概率选择下能够还原幅度，从而建立这两个概念之间的联系。