How Category Theory Works: The Elements & Distinctions Analysis of the Morphisms, Duality, and Universal Constructions in Sets

TL;DR

This paper demonstrates that elements and distinctions are fundamental concepts for analyzing morphisms, duality, and universal constructions in the category of sets, and extends these ideas to other structured categories and abstract category theory.

Contribution

It introduces a dual notions framework of elements and distinctions as core analytical tools for understanding category theory and extends these concepts to abstract categorical structures.

Findings

Elements and distinctions underpin morphism analysis in Sets

The framework extends to structured categories like groups and rings

Abstract arrow-theoretic definitions emerge from element and distinction concepts

Abstract

The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and functions. The analysis extends directly to other concrete categories (groups, rings, vector spaces, etc.) where the objects are sets with a certain type of structure and the morphisms are functions that preserve that structure. Then the elements & distinctions-based definitions can be abstracted in purely arrow-theoretic way for abstract category theory. In short, the language of elements & distinctions is the conceptual language in which the category of sets is written, and abstract category theory gives the abstract arrows version of those definitions.