Yoneda embedding

Definition

Let $\mathcal{C}$ be a locally small category . The Yoneda embedding is the functor

defined as follows:

• On objects $A\in \mathcal{C}$, set

  $$y(A) \;=\; \mathrm{Hom}_{\mathcal{C}}(-,A):\mathcal{C}^{op}\to \mathbf{Set}.$$

  This is a representable functor (a representable presheaf).

• On a morphism $f:A\to B$, define a natural transformation

  $$y(f):\mathrm{Hom}_{\mathcal{C}}(-,A)\Rightarrow \mathrm{Hom}_{\mathcal{C}}(-,B)$$

  by postcomposition:

  $$y(f)_X:\mathrm{Hom}_{\mathcal{C}}(X,A)\to \mathrm{Hom}_{\mathcal{C}}(X,B),\quad (h:X\to A)\mapsto f\circ h.$$

Here $\mathcal{C}^{op}$ is the opposite category and $\mathbf{Set}^{\mathcal{C}^{op}}$ is the functor category of presheaves on $\mathcal{C}$.

Fundamental property (fully faithful)

By the Yoneda lemma , the functor $y$ is fully faithful: for all objects $A,B\in\mathcal{C}$,

Equivalently, $\mathcal{C}$ identifies with a full subcategory of $\mathbf{Set}^{\mathcal{C}^{op}}$ whose objects are precisely the representables.

Examples

Example (Set)

For a set $A$, $y(A)$ is the presheaf

the set of functions $X\to A$. A map $A\to B$ induces a natural transformation by postcomposition.

Example (Posets)

Let $(P,\le)$ be a partial order regarded as a category (one morphism $x\to y$ iff $x\le y$).
Then for $a\in P$, the presheaf $y(a)$ sends $x$ to a singleton set if $x\le a$, and to the empty set otherwise. Thus $y(a)$ encodes the principal down-set $\{x\mid x\le a\}$.

Example (Grp)

For a group $G$, $y(G)$ is the presheaf

the set of group homomorphisms into $G$. A group homomorphism $G\to G'$ induces a natural transformation by postcomposition.