Some notes on diagram chasing and diagrammatic proofs in category theory

TL;DR

This paper discusses diagram chasing in category theory, contrasting element-based and diagrammatic approaches, and provides an original proof of the short five lemma using universal properties.

Contribution

It offers a clear exposition on diagram chasing, especially in abelian categories, and introduces a novel proof of the short five lemma via pullbacks.

Findings

Contrasts element-theoretic and diagrammatic methods in category theory.

Provides an original proof of the short five lemma using universal properties.

Highlights the importance of diagrammatic reasoning in understanding category theory.

Abstract

Diagram chasing is a customary proof method used in category theory and homological algebra. It involves an element-theoretic approach to show that certain properties hold for a commutative diagram. When dealing with abelian categories for the first time, one would work using a diagrammatic approach without relying on the notion of elements. However, constantly manipulating universal properties of various diagrams can be quite cumbersome. That said, we believe that it is still important to draw a contrast between both viewpoints in order to motivate the field of category theory. We focus our scope to the short five lemma, one of the more elementary diagram lemmas, and present a quick exposition on relevant subjects. Moreover, we give an original proof of the short five lemma using the universal property of pullbacks.