Adjoints of Morphisms of Neural Codes

TL;DR

This paper explores the structure of morphisms between neural codes, representing them with matrices, and introduces tools like adjoints and defect to analyze their properties and the induced partial order.

Contribution

It characterizes morphisms of neural codes using Galois connections and matrix factorizations, and introduces the concept of defect to study the code poset.

Findings

Morphisms form a Galois connection with boolean matrix multiplication.

Characterization of when morphisms allow boolean matrix factorization.

Introduction of the defect as a measure decreasing by 0 or 1 under coverings.

Abstract

A combinatorial code $\mathcal{C}$ is a collection of subsets of $[n]$, or equivalently a set of points in $\{0,1\}^n$. A morphism of codes is a map from one combinatorial code to another such that the coordinates of points in the image can be expressed as products of coordinates in the domain. By representing morphisms of codes as binary matrices, we show that any morphism of codes is part of a Galois connection where its adjoint is boolean multiplication by the representative matrix. We use this to characterize those morphisms of codes which allow to factor a boolean matrix, with applications to estimating boolean matrix rank. Morphisms also induce a partial order on (isomorphism classes of) codes. We determine the covering relations in this partial order for which the two adjoint maps are mutual inverses in terms of \emph{free} neurons, a combinatorial condition on the index corresponding to the covering maps. We introduce the \emph{defect} of a code as a new tool to study this poset and show that defect decreases by exactly 0 or 1 under a covering map.