nLab
Yoneda embedding

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 $\mathrm{Cat}$.

Hence $Y$ sends any object $c\in C$ to the presheaf which assigns to any other object $d$ of $c$ the 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 $S$.