Category Theory: Symmetry Group of Comma-propagation Transformations

TL;DR

This paper explores the symmetry group of comma-propagation transformations within category theory, revealing a hierarchical structure of arrow-categories and their invariance properties under a categorial-symmetry group.

Contribution

It introduces the concept of a categorial-symmetry group acting on infinite hierarchies of arrow-categories, extending the understanding of functorial constructions in algebraic topology.

Findings

Identification of a categorial-symmetry group $CS(\

) of comma-propagation transformations.

Hierarchical structure of arrow-categories with invariance under symmetry group actions.

Abstract

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. The arrow categories are more simple forms of the \emph{comma} categories and were introduced by Lawvere in the context of the interdefinability of the universal concepts of category theory. The basic idea is the elevation of arrows of one category $\textbf{C}$ to objects in another. Given a category (as a "geometric object") $\textbf{C}$ we can consider its properties (the universal categorial commutative diagrams) preserved under actions of a comma-propagation operation $\{\}$ in the infinite hierarchy of its arrow-categories (n-dimensional levels, such that for any $n\geq 1$, $\textbf{C}_{n+1} = {\textbf{C}_n}$, with $\textbf{C}_1 =\textbf{C}$) and on the functors (and their natural transformations) between such n-dimensional levels, which is a phenomena of a general categorial symmetry under a categorial-symmetry group $CS(\mathbb{Z})$ of all comma-propagation transformations.