Basic Set Theory and Algebra: Hints on Representation, Topology, Geometry, Analysis

TL;DR

This paper provides an accessible introduction to set theory, algebra, and category theory, then explores advanced topics like localization and homological algebra, linking these to geometry and analysis.

Contribution

It offers a comprehensive, low-prerequisite overview of advanced mathematical concepts and their interrelations, with practical hints for applications in geometry and analysis.

Findings

Clarifies foundational set and algebra concepts

Details category theory, limits, and abelian categories

Connects category theory to geometry and analysis

Abstract

In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat detailed discussion of system limits and the exact imbedding of abelian categories. This is followed (in part 3) by a discussion of localization, homological algebra, and generalizations of additive and abelian categories such as triangulated and derived categories. Based on the concepts of category theory from parts 2 and 3, (in part 4) we provide hints for constructive discussions on familiar mathematics such as representation theory and topological geometry/analysis (i.e., topology-based geometry/analysis). If events permit, the notes will be revised/updated regularly.