The Yoneda Lemma ([[米田引理]]), also referred to as the [[Yoneda Embedding]], is a fundamental result in category theory. It was discovered by the Japanese mathematician Nobuo Yoneda in the 1950s and has since become one of the key tools in the study of category theory. At its core, the Yoneda Lemma establishes a deep connection between objects in a category and certain functors that act on them. Specifically, it relates objects in a category to the set of all natural transformations from the hom-functor of that object to any other functor. The lemma states that for any object A in a category C, there is a one-to-one correspondence between morphisms from A to an object X and natural transformations from the hom-functor Hom(-,A) to Hom(-,X). In other words, morphisms from A to X can be fully characterized by natural transformations between two hom-functors. This result has far-reaching implications and provides powerful tools for understanding categories and their objects. It allows us to study objects through their relationships with other objects and functors. Moreover, it provides a way to represent objects by their morphisms rather than their internal structure. The Yoneda Lemma is widely used in various areas of mathematics. It plays a crucial role in algebraic geometry, algebraic topology, representation theory, and many other branches of mathematics. It helps establish connections between different mathematical structures and enables us to transport knowledge from one area to another. Furthermore, the Yoneda Lemma has deep philosophical implications. It suggests that understanding an object is equivalent to understanding all its relationships with other objects within its category. This idea aligns with the categorical viewpoint that focuses on relationships rather than internal structure. # Best Explanation so far For a systematic explanation, one can watch this video: by [[Andrius Kulikauskas]]  # Conclusion In summary, the Yoneda Lemma is a powerful tool that connects objects within a category through functors and natural transformations. It has wide-ranging applications in mathematics and provides insights into both the structure of categories and the nature of mathematical understanding. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Yoneda Lemma") or contains(subject, "Yoneda lemma") ```