Morphisms of Neural Codes

We define a notion of morphism between combinatorial codes, making the class of all combinatorial codes into a category Code. We show that morphisms can be used to remove redundant information from a code and that morphisms preserve convexity. We use the latter fact to define a partial order on all codes in which the class of convex codes forms a down-set. We investigate minimal obstructions to convexity in the form of "minimally nonconvex" codes, which lie on the boundary of this down-set. In particular, we show that there are infinitely many minimally nonconvex codes and construct a minimally nonconvex code with no local obstructions. We conclude by giving an algebraic formulation of our results.