Morphisms of Neural Codes
arXiv:1806.02014 · doi:10.1137/18M1205509
Abstract
We define a notion of morphism between combinatorial codes, making the class of all combinatorial codes into a category . We show that morphisms can be used to remove redundant information from a code, and that morphisms preserve convexity. This fact leads us to define "minimally non-convex" codes. We propose a program to characterize these minimal obstructions to convexity and hence characterize all convex codes. We implement a library of Sage code to perform computation with morphisms. These computational methods yield the smallest to-date example of a non-convex code with no local obstructions. We conclude by giving an algebraic formulation of our results.
Second version uploaded to correct an error in Remark 1.7, and to emphasize that the ambient space X must always be convex and open