Fibration symmetry-breaking supports functional transitions in a brain network engaged in language.

In his book 'A Beautiful Question' 1 , physicist Frank Wilczek argues that symmetry is 'nature's deep design,' governing the behavior of the universe, from the smallest particles to the largest structures 1-4 . While symmetry is a cornerstone of physics, it has not yet been found widespread applicability to describe biological systems 5 , particularly the human brain. In this context, we study the human brain network engaged in language and explore the relationship between the structural connectivity (connectome or structural network) and the emergent synchronization of the mesoscopic regions of interest (functional network). We explain this relationship through a different kind of symmetry than physical symmetry, derived from the categorical notion of Grothendieck fibrations 6 . This introduces a new understanding of the human brain by proposing a local symmetry theory of the connectome, which accounts for how the structure of the brain's network determines its coherent activity. Among the allowed patterns of structural connectivity, synchronization elicits different symmetry subsets according to the functional engagement of the brain. We show that the resting state is a particular realization of the cerebral synchronization pattern characterized by a fibration symmetry that is broken 7 in the transition from rest to language. Our findings suggest that the brain's network symmetry at the local level determines its coherent function, and we can understand this relationship from theoretical principles.