We share a small connection between information theory, algebra, and topology - namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster.

翻译：我们分享了信息理论、代数和地形学之间的一个小联系,即香农昆虫之间的对应关系,以及所演的表象。我们首先简要地回顾歌剧及其与表象性不一和真实线的表述,作为主要例子。然后我们给出了在任何类别中产生一个剧象的总定义,该剧团的数值在歌剧的双模体上都有。主要结果是香农昆虫定义了所演的表象性不一的衍生,对于这场歌剧的每一个衍生物,都存在一个由香农复方常数给出的点。我们展示了这一点与Faddeev在1956年给出的众所周知的昆虫特征以及Leinster最近给出的变异。