Entropy and the operad of simplices

This thesis explores a connection between information theory, algebra, and topology by studying the operadic structures of simplices and their relation to Shannon entropy. Beginning with classical notions of thermodynamic and Shannon entropy, we examine how probability distributions compose and how entropy behaves under such compositions. Using category theory and the framework of functors, we reinterpret Shannon entropy as a functor from the category of finite probability spaces to the additive monoid of nonnegative real numbers. Building upon this, we introduce the operad of simplices and show that the topological simplices naturally form an operad whose compositions mirror the combination of probability distributions. We then demonstrate that Shannon entropy defines a derivation on this operad, satisfying a generalised Leibniz rule. The thesis synthesises foundational results from Faddeev and recent developments by Leinster and Bradley, illustrating that Shannon entropy emerges as a universal and natural object in this operadic framework. This work reveals new algebraic and topological insights into the structural behaviour of entropy, opening pathways for further study into categorical approaches to information theory.