arXiv2025
A symmetry is a `change without change'. As simple as it sounds, this concept is the fundamental cornerstone that unifies all branches of theoretical physics. Virtually all physical laws -- ranging from classical mechanics and electrodynamics to relativity, quantum mechanics, and the standard model -- can be expressed in terms of symmetry invariances. In this book, we explore whether the same principle can also explain the emergent laws of biological systems. We introduce a new geometry for biological networks and AI architectures, drawing inspiration from the mystic genius of Grothendieck's fibrations in category theory. We attempt to bridge the gap between physics and biology using symmetries but with a twist. The traditional symmetry groups of physics are global and too rigid to describe biology. Instead, the novel notion of symmetry fibration is local, flexible, and adaptable to evolutionary pressures, providing the right framework for understanding biological complexity. In other words, this more general symmetry invariance is necessary and sufficient to ensure that a given biological network configuration can support a synchronized function. In this book, we review the theoretical progress over the last decades from mathematics, physics, computer science, dynamical systems, and graph theory that has led to the discovery of symmetry fibrations in biological networks. These symmetries act as organizing principles for biological networks. They serve as effective tools for describing the structure of these networks, blending geometry and topology. Fibrations explain how structure dictates function across various biological domains, including the transcriptome, proteome, metabolome, and connectome. Additionally, they facilitate a reduction in the dimensionality of the network, simplifying it into its fundamental building blocks for biological computation.