Representable functors are a powerful tool in category theory, connecting abstract categories to concrete sets. They're like translators, turning complex categorical structures into more familiar set-based ones.

The Yoneda embedding takes this idea further, showing how any category can be viewed as a category of set-valued functors. This perspective reveals hidden structures and relationships within categories, making it a cornerstone of advanced category theory.

Representable Functors

Examples of representable functors

• The forgetful functor $U: \mathbf{Grp} \to \mathbf{Set}$ is represented by the group $\mathbb{Z}$
  • $U$ maps each group to its underlying set and each group homomorphism to its underlying function
  • For any group $G$, there is a natural isomorphism between $\text{Hom}_{\mathbf{Grp}}(\mathbb{Z}, G)$ and $UG$, given by mapping a group homomorphism $f: \mathbb{Z} \to G$ to the element $f(1)$ in $G$
• The functor $\text{Hom}_{\mathbf{Vect}_k}(-, k): \mathbf{Vect}_k^{op} \to \mathbf{Set}$ is represented by the field $k$ as a vector space over itself
  • This functor maps each vector space to the set of linear maps from that space to $k$
  • For any vector space $V$, there is a natural isomorphism between $\text{Hom}_{\mathbf{Vect}k}(V, k)$ and $\text{Hom}{\mathbf{Vect}_k}(k, V)$, given by the transpose of a linear map
• For any object $A$ in a locally small category $\mathcal{C}$, the hom-functor $\text{Hom}_{\mathcal{C}}(-, A): \mathcal{C}^{op} \to \mathbf{Set}$ is representable by definition
  • This functor maps each object $X$ to the set $\text{Hom}{\mathcal{C}}(X, A)$ and each morphism $f: X \to Y$ to the function $\text{Hom}{\mathcal{C}}(f, A): \text{Hom}{\mathcal{C}}(Y, A) \to \text{Hom}{\mathcal{C}}(X, A)$ given by precomposition with $f$
  • The identity natural transformation on $\text{Hom}_{\mathcal{C}}(-, A)$ witnesses its representability by $A$

Yoneda Embedding

Construction of Yoneda embedding

• The Yoneda embedding is a functor $\mathcal{Y}: \mathcal{C} \to [\mathcal{C}^{op}, \mathbf{Set}]$ for a locally small category $\mathcal{C}$
  1. For each object $A$ in $\mathcal{C}$, $\mathcal{Y}(A)$ is defined as the hom-functor $\text{Hom}_{\mathcal{C}}(-, A): \mathcal{C}^{op} \to \mathbf{Set}$
     • This functor maps each object $X$ to the set $\text{Hom}{\mathcal{C}}(X, A)$ and each morphism $f: X \to Y$ to the function $\text{Hom}{\mathcal{C}}(f, A): \text{Hom}{\mathcal{C}}(Y, A) \to \text{Hom}{\mathcal{C}}(X, A)$ given by precomposition with $f$
  2. For each morphism $f: A \to B$ in $\mathcal{C}$, $\mathcal{Y}(f)$ is defined as the natural transformation $\text{Hom}{\mathcal{C}}(-, A) \to \text{Hom}{\mathcal{C}}(-, B)$ given by post-composition with $f$
     • For each object $X$ in $\mathcal{C}$, the component of $\mathcal{Y}(f)$ at $X$ is the function $\text{Hom}{\mathcal{C}}(X, f): \text{Hom}{\mathcal{C}}(X, A) \to \text{Hom}_{\mathcal{C}}(X, B)$ given by post-composition with $f$

Full faithfulness of Yoneda embedding

• To prove that the Yoneda embedding is fully faithful, we need to show that for any objects $A$ and $B$ in $\mathcal{C}$, the function $\mathcal{Y}{A,B}: \text{Hom}{\mathcal{C}}(A, B) \to \text{Hom}_{[\mathcal{C}^{op}, \mathbf{Set}]}(\mathcal{Y}A, \mathcal{Y}B)$ is a bijection
  1. Injectivity: If $\mathcal{Y}{A,B}(f) = \mathcal{Y}{A,B}(g)$ for morphisms $f, g: A \to B$, then $f = g$ by the Yoneda lemma
     • The Yoneda lemma states that for any natural transformation $\alpha: \text{Hom}{\mathcal{C}}(-, A) \to \text{Hom}{\mathcal{C}}(-, B)$, there exists a unique morphism $h: A \to B$ such that $\alpha = \mathcal{Y}_{A,B}(h)$, given by $h = \alpha_A(1_A)$
     • If $\mathcal{Y}{A,B}(f) = \mathcal{Y}{A,B}(g)$, then $f = \mathcal{Y}_{A,B}(f)A(1_A) = \mathcal{Y}{A,B}(g)_A(1_A) = g$
  2. Surjectivity: For any natural transformation $\alpha: \text{Hom}{\mathcal{C}}(-, A) \to \text{Hom}{\mathcal{C}}(-, B)$, there exists a unique morphism $f: A \to B$ such that $\alpha = \mathcal{Y}_{A,B}(f)$, given by $f = \alpha_A(1_A)$
     • This follows directly from the Yoneda lemma, as stated above

Representable functors vs Yoneda embedding

• The Yoneda embedding establishes a connection between representable functors and the category $[\mathcal{C}^{op}, \mathbf{Set}]$
  • Every representable functor $F: \mathcal{C}^{op} \to \mathbf{Set}$ is naturally isomorphic to $\mathcal{Y}A$ for some object $A$ in $\mathcal{C}$
    • If $F$ is represented by $A$, then there is a natural isomorphism $\alpha: F \to \mathcal{Y}A$ given by $\alpha_X(x) = \phi_X(x)$ for each object $X$, where $\phi: F \to \text{Hom}_{\mathcal{C}}(-, A)$ is the natural isomorphism witnessing the representability of $F$
  • Conversely, every functor in the image of the Yoneda embedding is representable
    • For any object $A$ in $\mathcal{C}$, the functor $\mathcal{Y}A = \text{Hom}_{\mathcal{C}}(-, A)$ is representable by $A$, as witnessed by the identity natural transformation on $\mathcal{Y}A$
• The Yoneda lemma states that for any functor $F: \mathcal{C}^{op} \to \mathbf{Set}$ and object $A$ in $\mathcal{C}$, there is a natural isomorphism $\text{Hom}_{[\mathcal{C}^{op}, \mathbf{Set}]}(\mathcal{Y}A, F) \cong FA$
  • This isomorphism is given by evaluating a natural transformation $\alpha: \mathcal{Y}A \to F$ at the identity morphism of $A$, i.e., mapping $\alpha$ to $\alpha_A(1_A)$
  • The inverse of this isomorphism maps an element $x \in FA$ to the natural transformation $\alpha: \mathcal{Y}A \to F$ defined by $\alpha_X(f) = Ff(x)$ for each object $X$ and morphism $f: X \to A$
  • The Yoneda lemma implies that the Yoneda embedding is fully faithful, as shown earlier
  • It also implies that representable functors are fully determined by their representing objects, since if $F$ is represented by $A$, then $F \cong \mathcal{Y}A$ and $A$ can be recovered (up to isomorphism) as the image of $1_A$ under the isomorphism $\text{Hom}_{[\mathcal{C}^{op}, \mathbf{Set}]}(\mathcal{Y}A, F) \cong FA$