A Categorical Development of Right Derived Functors

TL;DR

This paper constructs right derived functors within Abelian categories using purely categorical methods, avoiding set-theoretic or embedding theorems, thus providing a minimalist and abstract approach to homological algebra.

Contribution

It demonstrates that homological concepts like right derived functors can be developed entirely through categorical notions without relying on set-theoretic language or embedding theorems.

Findings

Develops cohomology groups from categorical sequences

Constructs right derived functors purely categorically

Provides a minimalist, abstract framework for homological algebra

Abstract

Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm of pure abstract category theory in their development of the field, leveraging the Freyd-Mitchell embedding theorem or similar results, or otherwise using set-theoretic language to augment a general categorical discussion. This paper seeks to demonstrate that - while it is not necessary for most mathematicians' purposes - a development of homological concepts can be contrived from purely categorical notions. We begin by outlining the categories we will work within, namely Abelian categories (building off additive categories). We continue to develop cohomology groups of sequences, eventually culminating in a development of right derived functors. This paper is designed to be a minimalist construction, supplying no examples or motivation beyond what is necessary to develop the ideas presented.