Author: Eiko

Tags: category theory, yoneda lemma

Time: 2024-11-26 14:39:07 - 2024-11-26 15:14:37 (UTC)

Yoneda’s Lemma

Let $h^X=\mathrm{Hom}(X,)$ and $h_Y=\mathrm{Hom}(,Y)$ be the covariant and contravariant $\mathrm{Hom}$ functors. The Yoneda’s Lemma states, in the category of (covariant and contravariant) functors, for a covariant functor $F:\mathcal{C}\to \mathbf{S}\mathbf{e}\mathbf{t}\mathbf{s}$ and contravariant functor $G:{\mathcal{C}}^{op}\to \mathbf{S}\mathbf{e}\mathbf{t}\mathbf{s}$, we have

• $\mathrm{Hom}(h^X,F)=F(X)$

• $\mathrm{Hom}(h_Y,G)=G(Y).$

As a special case we have

• $\mathrm{Hom}(h_X,h_Y)=\mathrm{Hom}(X,Y)$

• $\mathrm{Hom}(h^X,h^Y)=\mathrm{Hom}(Y,X)$

i.e. $h_{◻}:\mathcal{C}\to \mathrm{Hom}({\mathcal{C}}^{op},\mathbf{S}\mathbf{e}\mathbf{t}\mathbf{s})$ and $h^{◻}:{\mathcal{C}}^{op}\to \mathrm{Hom}(\mathcal{C},\mathbf{S}\mathbf{e}\mathbf{t}\mathbf{s})$ are fully faithful functors.