范畴幅度与熵
Categorical magnitude and entropy
AI总结：
本文将香农熵扩展到配备概率的有限范畴，建立范畴幅度与熵的联系，使得选取特定均匀概率时可恢复范畴幅度，实现基数的范畴论推广与信息论熵的统一。
摘要：
对于任意配备概率测度的有限集合，我们可以计算其香农熵或信息量。当采用均匀概率时，熵等于集合基数的对数。Leinster为某类有限范畴引入了欧拉示性数的概念，也称为幅度（magnitude），可视为基数的范畴论推广。本文旨在将这两个概念联系起来，考虑将香农熵扩展到配备概率的有限范畴，使得当选取某种“均匀”概率时，能够得到幅度。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
