Homotopical Approach to Tensor Products of Modules

TL;DR

This paper introduces a homotopical approach to tensor products of modules, aiming to make homotopical algebra more accessible by providing a self-contained, undergraduate-friendly guide to the fundamental concepts and results.

Contribution

It offers a comprehensive, accessible introduction to homotopical algebra related to tensor products, including a detailed discussion of Robinson's result and a computational proof of concept.

Findings

Homotopical methods relate torsion products to homotopy groups.

A self-contained guide simplifies complex categorical and algebraic concepts.

Demonstrates the approach with a computational example.

Abstract

Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the realization of torsion products as the homotopy groups of a topological space, which is itself constructed for this express purpose. However, even stating this result formally requires a multitude of preliminaries in algebra, topology, and category theory. The goal of this document is to present a self-contained guide to the fundamental concepts and results, with few proofs, required to do work with this kind of mathematics in hopes of making the field of homotopical algebra more accessible. We only assume familiarity with topological spaces and groups, so it is approachable from an undergraduate level. This project culminates in a discussion of the result of Robinson mentioned above along with a computation as a proof of concept.