A combinatorial neural code records which subsets of neurons fire together. The code captures the representational capacity of a neural population — which stimuli it can distinguish, which regions of stimulus space it can tile. Morphisms between codes formalize how one population's representation relates to another's.
Geraci, Kunin, and Seceleanu (arXiv:2603.10837) discover that these morphisms form a Galois connection. Each morphism between neural codes has a left and right adjoint, computed via boolean matrix multiplication. The adjoints are not approximations. They are exact — the tightest possible upper and lower bounds on how one code maps into another.
The “defect” — the gap between a morphism and its adjoint composition — quantifies how much information is lost or created in the mapping. Zero defect means the morphism is an embedding: the target code fully captures the source. Positive defect means the mapping introduces ambiguity: distinct representations in the source become indistinguishable in the target.
This is algebraic structure that nobody expected in neural coding theory. Galois connections appear in order theory, lattice theory, formal concept analysis — abstract mathematics with no obvious neural interpretation. But the structure is not imposed. It falls out of the combinatorics of which neurons fire together. The set of all possible firing patterns has a partial order, and morphisms between codes respect that order in a way that automatically generates adjoints.
The bridge between neural representation and abstract algebra is natural. The formalism doesn't interpret neural codes through algebraic lenses. It discovers that neural codes were algebraic objects all along — that the combinatorial constraints of population coding force Galois structure as a mathematical consequence.