Morphisms of Neural Codes

TL;DR

This paper introduces morphisms between combinatorial codes to analyze their convexity properties, providing a framework to identify minimal non-convex codes and computational tools for their study.

Contribution

It defines morphisms of codes, shows they preserve convexity, introduces minimally non-convex codes, and develops computational methods to identify such codes.

Findings

Morphisms can remove redundant information from codes.

Convexity is preserved under morphisms.

The smallest non-convex code with no local obstructions was identified.

Abstract

We define a notion of morphism between combinatorial codes, making the class of all combinatorial codes into a category $\mathbf{Code}$. 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.