# nLab restricted Yoneda embedding

Restricted Yoneda embedding

# 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, 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.

Last revised on February 16, 2021 at 14:10:22. See the history of this page for a list of all contributions to it.