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

• Max Kelly, sect. 1.9 (weak form) and 2.4 (strong from) of Basic Concepts of Enriched Category Theory, Cambridge University Press, Lecture Notes in Mathematics 64 (1982) (pdf)

• Francis Borceux, theorem 6.3.5 of Handbook of Categorical Algebra, Volume 2, Cambridge University Press (1994)

• Niles Johnson, Donald Yau, Bimonoidal Categories, $E_n$-Monoidal Categories, and Algebraic K-Theory (three volume book, Section III.3.7.

• Fosco Loregian, Coend calculus, Cambridge University Press 2021 (arXiv:1501.02503, doi:10.1017/9781108778657, ISBN:9781108778657), discusses the Set-enriched versions of the Yoneda lemma in Section 2.2, where the density forms are called the ninja Yoneda lemma.

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

• Ross Street, Skew-closed categories (arXiv:1205.6522)

• Vladimir Hinich, Enriched Yoneda lemma, Theory and Applications of Categories 31 29 (2016) 833-838 [tac:31-29, pdf]