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Neural category

Neha Gupta, Suhith K N

TL;DR

The main work in this paper comprises constructing two categories, the $\mathfrak{C}$ category, a subcategory of SETS consisting of neural codes and code maps and the neural category $\mathfrak{N}$, a subcategory of \textit{Rngs} consisting of neural rings and neural ring homomorphisms.

Abstract

A neural code on $ n $ neurons is a collection of subsets of the set $ [n]=\{1,2,\dots,n\} $. Curto et al. \cite{curto2013neural} associated a ring $\mathcal{R}_{\mathcal{C}}$ (neural ring) to a neural code $\mathcal{C}$. A special class of ring homomorphisms between two neural rings, called neural ring homomorphism, was introduced by Curto and Youngs \cite{curto2020neural}. The main work in this paper comprises constructing two categories. First is the $\mathfrak{C}$ category, a subcategory of SETS consisting of neural codes and code maps. Second is the neural category $\mathfrak{N}$, a subcategory of \textit{Rngs} consisting of neural rings and neural ring homomorphisms. Then, the rest of the paper characterizes the properties of these two categories like initial and final objects, products, coproducts, limits, etc. Also, we show that these two categories are in dual equivalence.

Neural category

TL;DR

The main work in this paper comprises constructing two categories, the category, a subcategory of SETS consisting of neural codes and code maps and the neural category , a subcategory of \textit{Rngs} consisting of neural rings and neural ring homomorphisms.

Abstract

A neural code on neurons is a collection of subsets of the set . Curto et al. \cite{curto2013neural} associated a ring (neural ring) to a neural code . A special class of ring homomorphisms between two neural rings, called neural ring homomorphism, was introduced by Curto and Youngs \cite{curto2020neural}. The main work in this paper comprises constructing two categories. First is the category, a subcategory of SETS consisting of neural codes and code maps. Second is the neural category , a subcategory of \textit{Rngs} consisting of neural rings and neural ring homomorphisms. Then, the rest of the paper characterizes the properties of these two categories like initial and final objects, products, coproducts, limits, etc. Also, we show that these two categories are in dual equivalence.
Paper Structure (3 sections, 15 theorems, 22 equations)

This paper contains 3 sections, 15 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. The neural category
  3. Further discussion

Key Result

Proposition 1

curto2020neural There is a 1-1 correspondence between code maps $q : \mathcal{C} \to \mathcal{D}$ and ring homomorphisms $\phi : \mathcal{R}_\mathcal{D} \to \mathcal{R}_\mathcal{C}$, given by the pullback map. That is, given a code map $q : \mathcal{C} \to \mathcal{D}$, its pullback $q^{\star} : \ma

Theorems & Definitions (35)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Definition 2
  • Lemma 1
  • Theorem 3
  • Definition 3
  • Example 1
  • Lemma 2
  • Lemma 3
  • ...and 25 more