The Yoneda embedding is a functor from a category C to its category of presheaves, $[C^{op}, Set]$. It is defined as follows: For any object $X$ in $C$, the Yoneda embedding associates the presheaf $h_X$, which is defined as follows: For any object $U$ in $C$, $h_X(U) = Hom(U, X)$. For any morphism $f: U$ -> $V$ in $C$, $h_X(f) = g$ -> $f * g$, where $g$ is any morphism in $Hom(V, X)$. **Yoneda Lemma** The Yoneda lemma is a fundamental result in category theory, which states that the Yoneda embedding is full and faithful. This means that it induces a natural bijection between the morphisms in $Hom(U, X)$ and the natural transformations between $h_U$ and $h_X$. ## Yoneda Embedding and Yoneda Lemma The Yoneda embedding and the Yoneda lemma are closely related. The Yoneda embedding provides a concrete way to represent objects in a category as [[presheaves]]. The Yoneda lemma then tells us that this representation is faithful, meaning that all of the information about an object is captured by its corresponding presheaf. **Applications of the Yoneda Lemma** The Yoneda lemma has many applications in category theory. For example, it can be used to prove the following: - Every functor from C to Set is naturally isomorphic to a representable functor. - Every limit and colimit in C is naturally isomorphic to a representable one. - Every universal construction in C is naturally isomorphic to a representable one. The Yoneda lemma is also a powerful tool for studying the relationship between different categories. For example, it can be used to prove the following: - The category of sheaves on a topological space is equivalent to the category of presheaves of sets on that space. - The category of abelian groups is equivalent to the category of chain complexes of modules over a principal ideal domain. The Yoneda lemma is a powerful and versatile result in category theory, with many applications in a variety of areas. # CHATGPT Answer In category theory, the Yoneda Embedding is a fundamental concept that relates to the Yoneda Lemma. The Yoneda Embedding is a functor that takes a category and embeds it into the category of presheaves on that category. To understand the Yoneda Embedding, let's first define what a presheaf is. In category theory, a presheaf on a category C is a contravariant functor from C to the category of sets. It assigns to each object in C a set, and to each morphism in C a function between sets. The Yoneda Embedding takes an object A in the original category C and maps it to the presheaf Hom(-, A), where Hom(-, A) is the contravariant hom-functor from C to sets. This means that for any object B in C, we have Hom(B, A) as the set of morphisms from B to A. The key insight of the Yoneda Embedding is that every object in C can be fully characterized by its relationship with other objects through these hom-sets. In other words, if we know all the morphisms from any object B to A for every object B in C, we can uniquely determine A up to isomorphism. This idea leads us to the Yoneda Lemma, which states that for any presheaf F on C and any object A in C, there is a natural bijection between natural transformations from Hom(-, A) to F and elements of F(A). This lemma establishes an important connection between objects in C and their corresponding presheaves. In summary, the Yoneda Embedding allows us to view objects in a category as presheaves on that category. The Yoneda Lemma then provides an essential tool for understanding and characterizing these presheaves based on their relationships with other objects. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Yoneda Lemma") or contains(subject, "Yoneda Embedding") ```