nLab
enriched Yoneda lemma

Idea

The enriched Yoneda lemma is the generalization of the usual Yoneda lemma from category theory to enriched category theory.

Statement

We discuss here two forms of the Yoneda lemma.

Let $V$ be a (locally small) closed symmetric monoidal category, so that $V$ is enriched in itself via its internal hom.

Weak form

A weak form of the enriched Yoneda lemma says that given a $V$-enriched functor $F:C\to V$ and an object $c$ of $C$, the set of $V$-enriched natural transformations $\alpha :{\mathrm{hom}}_C(c,-)\to F$ is in natural bijection with the set of elements of $F(c)$, i.e., the set of morphisms $I\to F(c)$, obtained by composition:

$$I\stackrel{1_c}{\to }{\mathrm{hom}}_C(c,c)\stackrel{\alpha c}{\to }F(c)$$I \stackrel{1_c}{\to} \hom_C(c, c) \stackrel{\alpha c}{\to} F(c)

Strong form

Now suppose that $V$ is in addition (small-)complete (has all small limits). Then, given a small $V$-enriched category $C$ and $V$-enriched functors $F,G:C\to V$, one may construct the object of $V$-natural transformations as an enriched end:

$$V^C(F,G)={\int }_{c}V(F(c),G(c))$$V^C(F, G) = \int_c V(F(c), G(c))

(which may in turn be expressed as an ordinary limit in $V$). This is the hom-object in the enriched functor category.

A strong form of the enriched Yoneda lemma specifies a $V$-natural isomorphism

$$V^C({\mathrm{hom}}_C(c,-),F)\cong F(c).$$V^C(\hom_C(c, -), F) \cong F(c).

This implies the weak form by applying the functor $\mathrm{hom}(I,-):V\to \mathrm{Set}$.

References

The weak form is in section 1.9, the strong form in section 2.4 of

• Max Kelly, Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982) (pdf)