Neural category

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.