nLab
Yoneda embedding

Idea

For $C$ a locally small category, every object $X$ of $C$ induces a presheaf on $C$: the representable presheaf $h_X$ represented by $X$. This assignment extends to a functor $C\to [C^{\mathrm{op}},\mathrm{Set}]$ from $C$ to its category of presheaves. The Yoneda lemma implies that this functor is full and faithful and hence realizes $C$ as a full subcategory inside its category of presheaves.

Recall from the discussion at representable presheaf that the presheaf represented by an object $X$ of $C$ is the functor $h_X:C^{\mathrm{op}}\to \mathrm{Set}$ whose assignment is illustrated by

which sends each object $U$ to ${\mathrm{Hom}}_C(U,X)$ and each morphism $\alpha :U\prime \to U$ to the function

$$h_X\alpha :{\mathrm{Hom}}_C(U,X)\to {\mathrm{Hom}}_C(U\prime ,X).$$h_X\alpha: Hom_C(U,X)\to Hom_C(U',X).

Moreover, for $f:X\to Y$ an morphism in $C$, this induces a natural transformation $h_f:h_X\to h_Y$, whose component on $U$ in $X$ is illustrated by

For this to be a natural transformation, we need to have the commuting diagram

$$\begin{array}{ccc}{h}_{X}U& \stackrel{h_{f}U}{\to }& h_{Y}U\\ h_X\alpha \downarrow & & \downarrow h_Y\alpha \\ h_{X}U\prime & \stackrel{h_{f}U\prime }{\to }& h_{Y}U\prime \end{array}$$\array{ h_X U & \stackrel{h_f U}{\rightarrow} & h_Y U \\ \mathllap{h_X\alpha\quad}{\downarrow} & {} & \mathrlap{\downarrow}{\quad h_Y\alpha} \\ h_X U' & \stackrel{h_f U'}{\rightarrow} & h_Y U' }

but this simply means that it doesn’t matter if we first “comb” the strands back to $U\prime$ and then comb the strands forward to $Y$, or comb the strands forward to $Y$ first and then comb the strands back to $U\prime$

which follows from associativity of composition of morphisms in $C$.

Definition

The Yoneda embedding for $C$ a locally small category is the functor

$$Y:C\to [C^{\mathrm{op}},\mathrm{Set}]$$Y : C \to [C^{op}, Set]

from $C$ to the category of presheaves over $C$ which is the image of the hom-functor

$$\mathrm{Hom}:C^{\mathrm{op}}\times C\to \mathrm{Set}$$Hom : C^{op} \times C \to Set

under the Hom adjunction

$$\mathrm{Hom}(C^{\mathrm{op}}\times C,\mathrm{Set})\simeq \mathrm{Hom}(C,[C^{\mathrm{op}},\mathrm{Set}])$$Hom(C^{op} \times C , Set) \simeq Hom(C, [C^{op}, Set])

in the closed symmetric monoidal category Cat.

Hence $Y$ sends any object $c\in C$ to the representable presheaf which assigns to any other object $d$ of $C$ the hom-set of morphisms from $d$ into $c$:

$$Y(c):C^{\mathrm{op}}\stackrel{C(-,c)}{\to }\mathrm{Set}.$$Y(c) : C^{op} \stackrel{C(-,c)}{\to} Set \,.

Properties

It follows from the Yoneda lemma that the functor $Y$ is full and faithful. It is also limit preserving (= continuous functor), but does in general not preserve colimits.

The Yoneda embedding of a small category $S$ into the category of presheaves on $S$ gives a free cocompletion of $S$.

If the Yoneda embedding of a category has a left adjoint, then that category is called a total category .