The Yoneda Lemma

Hi all! I realized I hadn’t talked about one of the most basic topics in category theory, which has been implicitly used in many of my posts, namely, the Yoneda Lemma. I’ll do so in this brief post.

If $\mathscr{C}$ is a category, then the Yoneda lemma says that the canonical functor $C\mapsto \mathrm{Hom}(-,C)$ is fully faithful. What is this functor? Any map $f:C\to D$ induces a map $\mathrm{Hom}(-C)\to \mathrm{Hom}(-,D)$ by taking any map $C^\prime\to C$ to $C^\prime\to C\xrightarrow{f} D$. The Yoneda lemma is the following statement.

Yoneda lemma: Let $X:\mathscr{C}^{op}\to \mathbf{Set}$ be a functor. There’s an isomorphism $\mathrm{Map}_{\mathrm{Fun}(\mathscr{C}^{op},\mathbf{Set})}(\mathrm{Hom}(-,C),X)\simeq X(C)$.

How do we prove this? Consider a natural transformation $\eta:\mathrm{Hom}(-,C)\to X$, and consider a functor $f:B\to C$. This induces maps $\mathrm{Hom}(C,C)\to \mathrm{Hom}(B,C)$, and maps $X(C)\to X(B)$, and there’s a commutative diagram:

Now, let’s consider what happens to the identity map $\mathrm{id}_C$ under this whole thing. We get:

However, because of commutativity, $\eta_b(f) = (X(f))(\eta_c(\mathrm{id}_C))$. So the transformation $\eta$ is determined by the map $f$. We therefore get the isomorphism in $\mathscr{C}$ given simply by taking $\mathrm{Map}_{\mathrm{Fun}(\mathscr{C}^{op},\mathbf{Set})}(\mathrm{Hom}(-,C),X)$ to $\mathrm{Map}_{\mathrm{Fun}(\mathscr{C}^{op},\mathbf{Set})}(\mathrm{Hom}(C,C),X(C))$, and then evaluating on $\mathrm{id}_C$. This gives $X(C)$, so we’re done!

The point of this result is that it makes precise the awesome philosophy of category theory that the properties of objects are determined by maps into that object. (There’s also a Yoneda lemma for higher categories, which states that if $X$ is a simplicial set, then the Yoneda embedding $X\to \mathscr{P}(X)$ is fully faithful.) Some examples of the Yoneda lemma in “real life” include the following (a very inexhaustive list):

1. In simplicial sets: if $S$ is a simplicial set, then $\mathrm{Hom}(\Delta^n,S)$ is simply $S_n$. This follows from the Yoneda lemma since $\Delta^n$ is the functor $\mathrm{Hom}(-,[n])$.
2. In algebraic topology: (singular, say) the $n$-th cohomology group with coefficients in a group $G$ is, by Brown representability, simply $\mathrm{Hom}(-,K(G,n))$. Therefore, natural transformations $\mathrm{H}^n(-,X)\to \mathrm{H}^m(-X)$ are maps $\mathrm{Hom}(-,K(G,n))\to\mathrm{Hom}(-,K(G,m))$, or, by the Yoneda lemma, maps $K(G,n)\to K(G,m)$. These are called cohomology operations.
3. If you’ve got a Hopf algebra, then its prime spectrum gives you an affine group scheme.

In general, the moral is that representing objects are unique up to isomorphism which could have similar structures as the functor. I hope this was fun!

Best,
SKD

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