The Yoneda Lemma¶

The Yoneda Lemma is a central result in Category Theory which states a natural isomorphism between the Hom-Set of natural transformationss from a Representable Functor to its representation and the image of the representing element under the Functor.

More explicitly, let $C$ be a locally-small category and $F$ be a set-valued functor on $C$. Then for any $c \in C$: $$ \text{Hom}(C(c, -), F) \cong Fc $$ is natural in both $F$ and $c$.

The Yoneda Embedding¶

A corollary to the Yoneda Lemma is the Yoneda embedding. Let $F: C \to \text{Set}^{C^{\text{op}}}$ such that $c \in C$ is mapped to $C(-, C)$. The Yoneda Lemma implies that this embedding is a fully faithful functor. A more colloquial way of stating this result is that natural transformations between represented functors correspond to morphisms between the representing objects.

Applications of the Yoneda Lemma¶

Row Operations on Matrices¶

Let $\text{Mat}_R$ be the category whose objects are positive integers and whose morphism $m \to n$ for $n, m \in \text{Mat}_R$ are $n \times m$ matrices whose entries are elements of the unital ring $R$. Composition is defined by matrix multiplication.

The elements in the image of $\text{Mat}_R(-, n)$ are matrices with $n$ rows. The standard row operations of linear algebra define natural endomorphisms of $\text{Mat}_R(-, n)$ (where naturality follows from linearity of matrix multiplication). Therefore, every row operation must be definable by left multiplication by a suitable $n \times n$ matrix. The Yoneda Lemma tells us that this matrix is the one gained by applying the row operations to the identity matrix.