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Internal Symmetry Group in Categorial Topology

Zoran Majkic

TL;DR

The paper addresses how symmetry can be internal to a category rather than global by formalizing Perfectly Symmetric Categories ($PSC$) through a duality $B_ op$ and developing the arrows-to-objects correspondence. It introduces a hierarchy of internal symmetries—Conceptually Closed Categories ($CoCC$), Symmetry-Extended Categories ($SEC$), and Imploded Categories ($IMC$)—via the closure functor $T_e$ and related structures, showing preservation of symmetry across all $n$-levels. The Internal Category-Symmetry Group $ICS(\mathbb{N})$ is defined as an infinite abelian group of endofunctor transformations generated by $E$, with an adjunction to the identity and a derived category $\widetilde{\mathbf{C}}$ that encodes conceptualized arrows; the group acts on objects and arrows and remains invariant under comma-propagation-like transformations. The work connects internal symmetries to global categorial symmetries, offering a universal, physics-informed perspective on symmetry in mathematics with potential links to Noether-type invariants and phase-like symmetries, while highlighting conditions under which PSCs yield stable, multi-level invariants.

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.

Internal Symmetry Group in Categorial Topology

TL;DR

The paper addresses how symmetry can be internal to a category rather than global by formalizing Perfectly Symmetric Categories () through a duality and developing the arrows-to-objects correspondence. It introduces a hierarchy of internal symmetries—Conceptually Closed Categories (), Symmetry-Extended Categories (), and Imploded Categories ()—via the closure functor and related structures, showing preservation of symmetry across all -levels. The Internal Category-Symmetry Group is defined as an infinite abelian group of endofunctor transformations generated by , with an adjunction to the identity and a derived category that encodes conceptualized arrows; the group acts on objects and arrows and remains invariant under comma-propagation-like transformations. The work connects internal symmetries to global categorial symmetries, offering a universal, physics-informed perspective on symmetry in mathematics with potential links to Noether-type invariants and phase-like symmetries, while highlighting conditions under which PSCs yield stable, multi-level invariants.

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 of all comma-propagation transformations), ia a local internal symmetry inside a given PSC category. Given a PSC category (as a "geometric object") we can consider its properties (the categorial commutative diagrams) preserved under actions of a particular endofunctor 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 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 ) . We show that if a base category is a PSC, then all its ne-dimensional levels are PSC as well.
Paper Structure (4 sections, 6 theorems, 31 equations)

This paper contains 4 sections, 6 theorems, 31 equations.

Table of Contents

  1. Introduction to Symmetry and Categorial Topology of n-dimensional Levels
  2. Introduction of the Hierarchy of Internal Symmetries
  3. Internal Symmetry Group of Transformations
  4. Conclusions

Key Result

Proposition 1

For each n-dimensional level $\textbf{C}_{n}$, $n\geq 2$, with its identity endofunctor $Id_{\textbf{C}_{n}}$, there exist the natural transformations $\sigma:\blacktriangle \circ F_{st}\rTo^\centerdot Id_{\textbf{C}_{n}}$ and $\sigma^{-1}:Id_{\textbf{C}_{n}}\rTo^\centerdot \blacktriangle\circ S_{nd so wit vertical composition $\sigma^{-1}\bullet\sigma = (\psi;\psi)$.

Theorems & Definitions (13)

  • definition 1
  • Proposition 1
  • definition 2
  • Proposition 2
  • Corollary 1
  • definition 3
  • definition 4
  • lemma 1
  • Corollary 2
  • definition 5
  • ...and 3 more