nLab restricted Yoneda embedding

Definition

For any functor $i:A\to C$ (often the inclusion of a full subcategory), the restricted Yoneda embedding is the composite

$$C \hookrightarrow [C^{op},Set] \xrightarrow{i^\ast} [A^{op},Set]$$

of the ordinary Yoneda embedding of $C$ with the restriction functor $i^\ast$ along $i$.

When $C$ is cocomplete and $A$ small, the restricted Yoneda embedding has a left adjoint: the realization.

One important example of a restricted Yoneda embedding is that of the fully faithful inclusion $i : \Delta \hookrightarrow Cat$, where $\Delta$ is the simplex category. This is known as the nerve functor.

Related pages

• Yoneda embedding
• dense subcategory
• nerve