Diagrammatic category theory

Abstract

In category theory, the use of string diagrams is well known to aid in the intuitive understanding of certain concepts, particularly when dealing with adjunctions and monoidal categories. We show that string diagrams are also useful in exploring fundamental properties of basic concepts in category theory, such as universal properties, (co)limits, Kan extensions, and (co)ends. For instance, string diagrams are utilized to represent visually intuitive proofs of the Yoneda lemma, necessary and sufficient conditions for being adjunctions, the fact that right adjoints preserve limits (RAPL), and necessary and sufficient conditions for having pointwise Kan extensions. We also introduce a method for intuitively calculating (co)ends using diagrammatic representations and employ it to prove several properties of (co)ends and weighted (co)limits. This paper proposes that using string diagrams is an effective approach for beginners in category theory to learn the fundamentals of the subject in an intuitive and understandable way.