Yoneda embedding - Imported Knowlpedia Content

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

y:\mathcal{C}\longrightarrow \mathbf{Set}^{\mathcal{C}^{op}}

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 $A,B\in\mathcal{C}$ ,

\mathrm{Hom}_{\mathcal{C}}(A,B)\;\cong\;\mathrm{Nat}\big(\mathrm{Hom}_{\mathcal{C}}(-,A),\,\mathrm{Hom}_{\mathcal{C}}(-,B)\big).

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

X \longmapsto \mathrm{Hom}_{\mathbf{Set}}(X,A),

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

H \longmapsto \mathrm{Hom}_{\mathbf{Grp}}(H,G),

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