nLab restricted Yoneda embedding

Restricted Yoneda embedding

Restricted Yoneda embedding


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

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

of the ordinary Yoneda embedding of CC with the restriction functor i *i^\ast along ii.

When CC 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:ΔCati : \Delta \hookrightarrow Cat, where Δ\Delta is the simplex category. This is known as the nerve functor.

Last revised on February 16, 2021 at 19:10:22. See the history of this page for a list of all contributions to it.