Adjoints of Morphisms of Neural Codes

TL;DR

The \emph{defect} of a code is introduced as a new tool to study this poset and it is shown that defect decreases by exactly 0 or 1 under a covering map.

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.

Figures (2)

• Figure 1: The Hasse diagram of the downset in $\mathbf{P_{Code}}$ generated by all codes of minimum neuron number 3. Each gray grid square contains all codes with that combination of trunk number and defect. Solid lines indicate covering relations which are matrix factorizations.
• Figure 2: The subset of edges in \ref{['fig:pcode3']} that correspond to matrix factorizations.

Theorems & Definitions (88)

• Example 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 2.5
• Remark 2.6
• Lemma 2.7
• Proposition 3.1: curto2019algebraic*Proposition 3.7
• Proposition 3.2
• proof : Proof of \ref{['lem:intgens', 'lem:uniongens']}