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The enriched Yoneda lemma is the generalization of the usual Yoneda lemma from category theory to enriched category theory.
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.
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:
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:
(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
This implies the weak form by applying the functor $\hom(I, -): V \to Set$.
If $V$ is also (small-)cocomplete, there is an equivalent dual formulation of the enriched Yondeda lemma as a $V$-natural isomorphism
induced by the morphisms
adjoint to $F_{c,d}$. This version is sometimes called the density theorem and sometimes simply called the enriched Yoneda lemma.
For an enriched presheaf $H: C^{op} \to V$ there are dual forms obtained by replacing $C$ above with $C^{op}$. These are
and
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]
Last revised on May 3, 2023 at 10:48:35. See the history of this page for a list of all contributions to it.