Title: A Compositional Approach to Higher-Order Structure in Complex Systems: Carving Nature at its Joints

Abstract: Relating microscopic interactions to macroscopic observables is a central challenge in the study of complex systems. Addressing this question requires understanding both pairwise and higher-order interactions, but the latter are less well understood. Here, we show that the M\"obius inversion theorem provides a general mathematical formalism for deriving higher-order interactions from macroscopic observables, relative to a chosen decomposition of the system into parts. Applying this framework to a diverse range of systems, we demonstrate that many existing notions of higher-order interactions, from epistasis in genetics and many-body couplings in physics, to synergy in game theory and artificial intelligence, naturally arise from an appropriate mereological decomposition. By revealing the common mathematical structure underlying seemingly disparate phenomena, our work highlights the fundamental role of decomposition choice in the definition and estimation of higher-order interactions. We discuss how this unifying perspective can facilitate the transfer of insights between domains, guide the selection of appropriate system decompositions, and motivate the search for novel interaction types through creative decomposition strategies. More broadly, our results suggest that the M\"obius inversion theorem provides a powerful lens for understanding the emergence of complex behaviour from the interplay of microscopic parts, with applications across a wide range of disciplines.