Symmetry plays a crucial role across various scales in physics-from the fundamental particles that comprise matter to the intricate shapes of snowflakes. The ubiquity of these symmetries poses a pivotal question: if life arises as an emergent property from physics, what prevents symmetry from also explaining the architectures of biological, or even artificial, life? This book addresses the question by introducing a new geometry for 'living' networks, drawing inspiration from Grothendieck's fibrations in category theory. The traditional, restrictive symmetry groups of physics are replaced with symmetry fibration, a novel notion which is both local and adaptable to evolutionary pressures. This provides an effective framework for understanding biological complexity, translating the once inscrutable AI 'black box' into an interpretable 'colored box' rich with symmetry. Featuring numerous cutting-edge applications from genomics, neuroscience and AI, this text is ideal for graduate students and researchers in mathematical biology, machine learning and network science.
