Aspects of algebraic exponentiation

TL;DR

This paper investigates algebraic exponentiation in the category of groups, revealing new universal properties and offering a novel approach to the centralization problem in categorical algebra.

Contribution

It introduces a new perspective on algebraic exponentiation and explores its implications for universal properties and centralization in group theory.

Findings

Algebraic exponentiation provides unique universal properties in groups.

The notion offers a new approach to the centralization problem.

Various properties of the category of groups are derived from this notion.

Abstract

We analyse some aspects of the notion of algebraic exponentiation introduced by the second author [16] and satisfied by the category of groups. We show how this notion provides a new approach to the categorical-algebraic question of the centralization. We explore, in the category of groups, the unusual universal properties and constructions determined by this notion, and we show how it is the origin of various properties of this category.