Internal Symmetry Group in Categorial Topology

TL;DR

This paper introduces the concept of internal categorial symmetry within Perfectly Symmetric Categories (PSC), revealing how local internal symmetries relate to global categorial symmetries across different levels.

Contribution

It defines and analyzes the internal categorial symmetry group in PSC, establishing relationships between local internal symmetries and global symmetries across n-dimensional levels.

Findings

Internal categorial invariance under endofunctor actions

Existence of an internal symmetry group $ICS( extbf{N})$

PSC property preserved across all n-dimensional levels

Abstract

The interdefinability of the universal concepts of category theory has been introduced by Lawvere. The perfect interdefinability between the objects and arrows of some category, defines the class of Perfectly Symmetric Categories (PSC) where each category can be represented equivalently by its arrows or by its objects only. Such symmetry, differently from the global categorial symmetry ( categorial-symmetry group $CS(\mathbb{Z})$ of all comma-propagation transformations), ia a local internal symmetry inside a given PSC category. Given a PSC category (as a "geometric object") $\textbf{C}$ we can consider its properties (the categorial commutative diagrams) preserved under actions of a particular endofunctor $E$ which transforms any commutative diagram into an invariant "up to isomorphism" diagram. We show that this kind of internal categorial invariance is a phenomena of a local categorial symmetry under an Internal Catergorial Symmetry group $ICS(\mathbb{N})$ of all local enfdofunctorial transformations. Then we establish the relationships between this local internal symmetry and global general symmetry between n-dimensional levels (the comma categories obtained from a PSC category $\textbf{C}$) . We show that if a base category $\textbf{C}$ is a PSC, then all its ne-dimensional levels are PSC as well.