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category theory

Yoneda lemma

# Contents

## 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 $\alpha: \hom_C(c, -) \to F$ 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:

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

### 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:

$V^C(F, G) = \int_c V(F(c), G(c))$

(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

$V^C(\hom_C(d, -), F) = \int_c V(\hom_C(d,c),F(c)) \cong F(d).$

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

### Density form

If $V$ is also (small-)cocomplete, there is an equivalent dual formulation of the enriched Yondeda lemma as a $V$-natural isomorphism

$\int^c \hom_C(c,d) \otimes F(c) \cong F(d)$

induced by the morphisms

$\hom_C(c,d) \otimes F(c) \to F(d)$

adjoint to $F_{c,d}$. This version is sometimes called the density theorem and sometimes simply called the enriched Yoneda lemma.

### Dual forms

For an enriched presheaf $H: C^{op} \to V$ there are dual forms obtained by replacing $C$ above with $C^{op}$. These are

$\int_c V(\hom_C(c,d),H(c)) \cong H(d)$

and

$\int^c \hom_C(d,c) \otimes H(c) \cong H(d).$

## References

• Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 (pdf)

Textbook accounts include

Generalizations to the case that the enriching monoidal category is not closed or symmetric (using skew-symmetric categories? or tensored categories?) can be found in

Last revised on May 3, 2023 at 10:48:35. See the history of this page for a list of all contributions to it.